import java.io.*;
import java.math.BigInteger;
import java.util.*;
import java.util.function.BiConsumer;
import java.util.function.Consumer;
import java.util.function.Function;
import java.util.function.Supplier;
import java.util.stream.Stream;

public class C_467
{
	public static final long[] POWER2 = generatePOWER2();
	public static final IteratorBuffer<Long> ITERATOR_BUFFER_PRIME = new IteratorBuffer<>(streamPrime(1000000).iterator());
	public static long BIG = 1000000000 + 7;
	private static BufferedReader bufferedReader = new BufferedReader(new InputStreamReader(System.in));
	private static StringTokenizer stringTokenizer = null;
	private static PrintWriter out = new PrintWriter(new BufferedOutputStream(System.out));

	static class Array<Type> implements Iterable<Type>
	{
		private final Object[] array;

		public Array(int size)
		{
			this.array = new Object[size];
		}

		public Array(int size, Type element)
		{
			this(size);
			Arrays.fill(this.array, element);
		}

		public Array(Array<Type> array, Type element)
		{
			this(array.size() + 1);
			for (int index = 0; index < array.size(); index++)
			{
				set(index, array.get(index));
			}
			set(size() - 1, element);
		}

		public Array(List<Type> list)
		{
			this(list.size());
			int index = 0;
			for (Type element : list)
			{
				set(index, element);
				index += 1;
			}
		}

		public Type get(int index)
		{
			return (Type) this.array[index];
		}

		@Override
		public Iterator<Type> iterator()
		{
			return new Iterator<Type>()
			{
				int index = 0;

				@Override
				public boolean hasNext()
				{
					return this.index < size();
				}

				@Override
				public Type next()
				{
					Type result = Array.this.get(index);
					index += 1;
					return result;
				}
			};
		}

		public Array set(int index, Type value)
		{
			this.array[index] = value;
			return this;
		}

		public int size()
		{
			return this.array.length;
		}

		public List<Type> toList()
		{
			List<Type> result = new ArrayList<>();
			for (Type element : this)
			{
				result.add(element);
			}
			return result;
		}

		@Override
		public String toString()
		{
			return "[" + C_467.toString(this, ", ") + "]";
		}
	}

	static class BIT
	{
		private static int lastBit(int index)
		{
			return index & -index;
		}

		private final long[] tree;

		public BIT(int size)
		{
			this.tree = new long[size];
		}

		public void add(int index, long delta)
		{
			index += 1;
			while (index <= this.tree.length)
			{
				tree[index - 1] += delta;
				index += lastBit(index);
			}
		}

		public long prefix(int end)
		{
			long result = 0;
			while (end > 0)
			{
				result += this.tree[end - 1];
				end -= lastBit(end);
			}
			return result;
		}

		public int size()
		{
			return this.tree.length;
		}

		public long sum(int start, int end)
		{
			return prefix(end) - prefix(start);
		}
	}

	static abstract class Edge<TypeVertex extends Vertex<TypeVertex, TypeEdge>, TypeEdge extends Edge<TypeVertex, TypeEdge>>
	{
		public final TypeVertex vertex0;
		public final TypeVertex vertex1;
		public final boolean bidirectional;

		public Edge(TypeVertex vertex0, TypeVertex vertex1, boolean bidirectional)
		{
			this.vertex0 = vertex0;
			this.vertex1 = vertex1;
			this.bidirectional = bidirectional;
			this.vertex0.edges.add(getThis());
			if (this.bidirectional)
			{
				this.vertex1.edges.add(getThis());
			}
		}

		public abstract TypeEdge getThis();

		public TypeVertex other(Vertex<TypeVertex, TypeEdge> vertex)
		{
			TypeVertex result;
			if (vertex0 == vertex)
			{
				result = vertex1;
			}
			else
			{
				result = vertex0;
			}
			return result;
		}

		public void remove()
		{
			this.vertex0.edges.remove(getThis());
			if (this.bidirectional)
			{
				this.vertex1.edges.remove(getThis());
			}
		}

		@Override
		public String toString()
		{
			return this.vertex0 + "->" + this.vertex1;
		}
	}

	public static class EdgeDefault<TypeVertex extends Vertex<TypeVertex, EdgeDefault<TypeVertex>>> extends Edge<TypeVertex, EdgeDefault<TypeVertex>>
	{
		public EdgeDefault(TypeVertex vertex0, TypeVertex vertex1, boolean bidirectional)
		{
			super(vertex0, vertex1, bidirectional);
		}

		@Override
		public EdgeDefault<TypeVertex> getThis()
		{
			return this;
		}
	}

	public static class EdgeDefaultDefault extends Edge<VertexDefaultDefault, EdgeDefaultDefault>
	{
		public EdgeDefaultDefault(VertexDefaultDefault vertex0, VertexDefaultDefault vertex1, boolean bidirectional)
		{
			super(vertex0, vertex1, bidirectional);
		}

		@Override
		public EdgeDefaultDefault getThis()
		{
			return this;
		}
	}

	public static class FIFO<Type>
	{
		public SingleLinkedList<Type> start;
		public SingleLinkedList<Type> end;

		public FIFO()
		{
			this.start = null;
			this.end = null;
		}

		public boolean isEmpty()
		{
			return this.start == null;
		}

		public Type peek()
		{
			return this.start.element;
		}

		public Type pop()
		{
			Type result = this.start.element;
			this.start = this.start.next;
			return result;
		}

		public void push(Type element)
		{
			SingleLinkedList<Type> list = new SingleLinkedList<>(element, null);
			if (this.start == null)
			{
				this.start = list;
				this.end = list;
			}
			else
			{
				this.end.next = list;
				this.end = list;
			}
		}
	}

	static class Fraction implements Comparable<Fraction>
	{
		public static final Fraction ZERO = new Fraction(0, 1);

		public static Fraction fraction(long whole)
		{
			return fraction(whole, 1);
		}

		public static Fraction fraction(long numerator, long denominator)
		{
			Fraction result;
			if (denominator == 0)
			{
				throw new ArithmeticException();
			}
			if (numerator == 0)
			{
				result = Fraction.ZERO;
			}
			else
			{
				int sign;
				if (numerator < 0 ^ denominator < 0)
				{
					sign = -1;
					numerator = Math.abs(numerator);
					denominator = Math.abs(denominator);
				}
				else
				{
					sign = 1;
				}
				long gcd = gcd(numerator, denominator);
				result = new Fraction(sign * numerator / gcd, denominator / gcd);
			}
			return result;
		}

		public final long numerator;
		public final long denominator;

		private Fraction(long numerator, long denominator)
		{
			this.numerator = numerator;
			this.denominator = denominator;
		}

		public Fraction add(Fraction fraction)
		{
			return fraction(this.numerator * fraction.denominator + fraction.numerator * this.denominator, this.denominator * fraction.denominator);
		}

		@Override
		public int compareTo(Fraction that)
		{
			return Long.compare(this.numerator * that.denominator, that.numerator * this.denominator);
		}

		public Fraction divide(Fraction fraction)
		{
			return multiply(fraction.inverse());
		}

		public boolean equals(Fraction that)
		{
			return this.compareTo(that) == 0;
		}

		public boolean equals(Object that)
		{
			return this.compareTo((Fraction) that) == 0;
		}

		public Fraction getRemainder()
		{
			return fraction(this.numerator - getWholePart() * denominator, denominator);
		}

		public long getWholePart()
		{
			return this.numerator / this.denominator;
		}

		public Fraction inverse()
		{
			return fraction(this.denominator, this.numerator);
		}

		public Fraction multiply(Fraction fraction)
		{
			return fraction(this.numerator * fraction.numerator, this.denominator * fraction.denominator);
		}

		public Fraction neg()
		{
			return fraction(-this.numerator, this.denominator);
		}

		public Fraction sub(Fraction fraction)
		{
			return add(fraction.neg());
		}

		@Override
		public String toString()
		{
			String result;
			if (getRemainder().equals(Fraction.ZERO))
			{
				result = "" + this.numerator;
			}
			else
			{
				result = this.numerator + "/" + this.denominator;
			}
			return result;
		}
	}

	static class IteratorBuffer<Type>
	{
		private Iterator<Type> iterator;
		private List<Type> list;

		public IteratorBuffer(Iterator<Type> iterator)
		{
			this.iterator = iterator;
			this.list = new ArrayList<Type>();
		}

		public Iterator<Type> iterator()
		{
			return new Iterator<Type>()
			{
				int index = 0;

				@Override
				public boolean hasNext()
				{
					return this.index < list.size() || IteratorBuffer.this.iterator.hasNext();
				}

				@Override
				public Type next()
				{
					if (list.size() <= this.index)
					{
						list.add(iterator.next());
					}
					Type result = list.get(index);
					index += 1;
					return result;
				}
			};
		}
	}

	public static class MapCount<Type> extends SortedMapAVL<Type, Long>
	{
		private int count;

		public MapCount(Comparator<? super Type> comparator)
		{
			super(comparator);
			this.count = 0;
		}

		public long add(Type key, Long delta)
		{
			long result;
			if (delta > 0)
			{
				Long value = get(key);
				if (value == null)
				{
					value = delta;
				}
				else
				{
					value += delta;
				}
				put(key, value);
				result = delta;
			}
			else
			{
				result = 0;
			}
			this.count += result;
			return result;
		}

		public int count()
		{
			return this.count;
		}

		public List<Type> flatten()
		{
			List<Type> result = new ArrayList<>();
			for (Entry<Type, Long> entry : entrySet())
			{
				for (long index = 0; index < entry.getValue(); index++)
				{
					result.add(entry.getKey());
				}
			}
			return result;
		}

		@Override
		public SortedMapAVL<Type, Long> headMap(Type keyEnd)
		{
			throw new UnsupportedOperationException();
		}

		@Override
		public void putAll(Map<? extends Type, ? extends Long> map)
		{
			throw new UnsupportedOperationException();
		}

		public long remove(Type key, Long delta)
		{
			long result;
			if (delta > 0)
			{
				Long value = get(key) - delta;
				if (value <= 0)
				{
					result = delta + value;
					remove(key);
				}
				else
				{
					result = delta;
					put(key, value);
				}
			}
			else
			{
				result = 0;
			}
			this.count -= result;
			return result;
		}

		@Override
		public Long remove(Object key)
		{
			Long result = super.remove(key);
			this.count -= result;
			return result;
		}

		@Override
		public SortedMapAVL<Type, Long> subMap(Type keyStart, Type keyEnd)
		{
			throw new UnsupportedOperationException();
		}

		@Override
		public SortedMapAVL<Type, Long> tailMap(Type keyStart)
		{
			throw new UnsupportedOperationException();
		}
	}

	public static class MapSet<TypeKey, TypeValue> extends SortedMapAVL<TypeKey, SortedSetAVL<TypeValue>> implements Iterable<TypeValue>
	{
		private Comparator<? super TypeValue> comparatorValue;

		public MapSet(Comparator<? super TypeKey> comparatorKey, Comparator<? super TypeValue> comparatorValue)
		{
			super(comparatorKey);
			this.comparatorValue = comparatorValue;
		}

		public MapSet(Comparator<? super TypeKey> comparatorKey, SortedSetAVL<Entry<TypeKey, SortedSetAVL<TypeValue>>> entrySet, Comparator<? super TypeValue> comparatorValue)
		{
			super(comparatorKey, entrySet);
			this.comparatorValue = comparatorValue;
		}

		public boolean add(TypeKey key, TypeValue value)
		{
			SortedSetAVL<TypeValue> set = computeIfAbsent(key, k -> new SortedSetAVL<>(comparatorValue));
			return set.add(value);
		}

		public TypeValue firstValue()
		{
			TypeValue result;
			Entry<TypeKey, SortedSetAVL<TypeValue>> firstEntry = firstEntry();
			if (firstEntry == null)
			{
				result = null;
			}
			else
			{
				result = firstEntry.getValue().first();
			}
			return result;
		}

		@Override
		public MapSet<TypeKey, TypeValue> headMap(TypeKey keyEnd)
		{
			return new MapSet<>(this.comparator, this.entrySet.headSet(new AbstractMap.SimpleEntry<>(keyEnd, null)), this.comparatorValue);
		}

		public Iterator<TypeValue> iterator()
		{
			return new Iterator<TypeValue>()
			{
				Iterator<SortedSetAVL<TypeValue>> iteratorValues = values().iterator();
				Iterator<TypeValue> iteratorValue = null;

				@Override
				public boolean hasNext()
				{
					return iteratorValues.hasNext() || (iteratorValue != null && iteratorValue.hasNext());
				}

				@Override
				public TypeValue next()
				{
					if (iteratorValue == null || !iteratorValue.hasNext())
					{
						iteratorValue = iteratorValues.next().iterator();
					}
					return iteratorValue.next();
				}
			};
		}

		public TypeValue lastValue()
		{
			TypeValue result;
			Entry<TypeKey, SortedSetAVL<TypeValue>> lastEntry = lastEntry();
			if (lastEntry == null)
			{
				result = null;
			}
			else
			{
				result = lastEntry.getValue().last();
			}
			return result;
		}

		public boolean removeSet(TypeKey key, TypeValue value)
		{
			boolean result;
			SortedSetAVL<TypeValue> set = get(key);
			if (set == null)
			{
				result = false;
			}
			else
			{
				result = set.remove(value);
				if (set.size() == 0)
				{
					remove(key);
				}
			}
			return result;
		}

		@Override
		public MapSet<TypeKey, TypeValue> tailMap(TypeKey keyStart)
		{
			return new MapSet<>(this.comparator, this.entrySet.tailSet(new AbstractMap.SimpleEntry<>(keyStart, null)), this.comparatorValue);
		}
	}

	public static class Matrix
	{
		public final int rows;
		public final int columns;
		public final Fraction[][] cells;

		public Matrix(int rows, int columns)
		{
			this.rows = rows;
			this.columns = columns;
			this.cells = new Fraction[rows][columns];
			for (int row = 0; row < rows; row++)
			{
				for (int column = 0; column < columns; column++)
				{
					set(row, column, Fraction.ZERO);
				}
			}
		}

		public void add(int rowSource, int rowTarget, Fraction fraction)
		{
			for (int column = 0; column < columns; column++)
			{
				this.cells[rowTarget][column] = this.cells[rowTarget][column].add(this.cells[rowSource][column].multiply(fraction));
			}
		}

		private int columnPivot(int row)
		{
			int result = this.columns;
			for (int column = this.columns - 1; 0 <= column; column--)
			{
				if (this.cells[row][column].compareTo(Fraction.ZERO) != 0)
				{
					result = column;
				}
			}
			return result;
		}

		public void reduce()
		{
			for (int rowMinimum = 0; rowMinimum < this.rows; rowMinimum++)
			{
				int rowPivot = rowPivot(rowMinimum);
				if (rowPivot != -1)
				{
					int columnPivot = columnPivot(rowPivot);
					Fraction current = this.cells[rowMinimum][columnPivot];
					Fraction pivot = this.cells[rowPivot][columnPivot];
					Fraction fraction = pivot.inverse().sub(current.divide(pivot));
					add(rowPivot, rowMinimum, fraction);
					for (int row = rowMinimum + 1; row < this.rows; row++)
					{
						if (columnPivot(row) == columnPivot)
						{
							add(rowMinimum, row, this.cells[row][columnPivot(row)].neg());
						}
					}
				}
			}
		}

		private int rowPivot(int rowMinimum)
		{
			int result = -1;
			int pivotColumnMinimum = this.columns;
			for (int row = rowMinimum; row < this.rows; row++)
			{
				int pivotColumn = columnPivot(row);
				if (pivotColumn < pivotColumnMinimum)
				{
					result = row;
					pivotColumnMinimum = pivotColumn;
				}
			}
			return result;
		}

		public void set(int row, int column, Fraction value)
		{
			this.cells[row][column] = value;
		}

		public String toString()
		{
			String result = "";
			for (int row = 0; row < rows; row++)
			{
				for (int column = 0; column < columns; column++)
				{
					result += this.cells[row][column] + "\t";
				}
				result += "\n";
			}
			return result;
		}
	}

	public static class Node<Type>
	{
		public static <Type> Node<Type> balance(Node<Type> result)
		{
			while (result != null && 1 < Math.abs(height(result.left) - height(result.right)))
			{
				if (height(result.left) < height(result.right))
				{
					Node<Type> right = result.right;
					if (height(right.right) < height(right.left))
					{
						result = new Node<>(result.value, result.left, right.rotateRight());
					}
					result = result.rotateLeft();
				}
				else
				{
					Node<Type> left = result.left;
					if (height(left.left) < height(left.right))
					{
						result = new Node<>(result.value, left.rotateLeft(), result.right);
					}
					result = result.rotateRight();
				}
			}
			return result;
		}

		public static <Type> Node<Type> clone(Node<Type> result)
		{
			if (result != null)
			{
				result = new Node<>(result.value, clone(result.left), clone(result.right));
			}
			return result;
		}

		public static <Type> Node<Type> delete(Node<Type> node, Type value, Comparator<? super Type> comparator)
		{
			Node<Type> result;
			if (node == null)
			{
				result = null;
			}
			else
			{
				int compare = comparator.compare(value, node.value);
				if (compare == 0)
				{
					if (node.left == null)
					{
						result = node.right;
					}
					else
					{
						if (node.right == null)
						{
							result = node.left;
						}
						else
						{
							Node<Type> first = first(node.right);
							result = new Node<>(first.value, node.left, delete(node.right, first.value, comparator));
						}
					}
				}
				else
				{
					if (compare < 0)
					{
						result = new Node<>(node.value, delete(node.left, value, comparator), node.right);
					}
					else
					{
						result = new Node<>(node.value, node.left, delete(node.right, value, comparator));
					}
				}
				result = balance(result);
			}
			return result;
		}

		public static <Type> Node<Type> first(Node<Type> result)
		{
			while (result.left != null)
			{
				result = result.left;
			}
			return result;
		}

		public static <Type> Node<Type> get(Node<Type> node, Type value, Comparator<? super Type> comparator)
		{
			Node<Type> result;
			if (node == null)
			{
				result = null;
			}
			else
			{
				int compare = comparator.compare(value, node.value);
				if (compare == 0)
				{
					result = node;
				}
				else
				{
					if (compare < 0)
					{
						result = get(node.left, value, comparator);
					}
					else
					{
						result = get(node.right, value, comparator);
					}
				}
			}
			return result;
		}

		public static <Type> Node<Type> head(Node<Type> node, Type value, Comparator<? super Type> comparator)
		{
			Node<Type> result;
			if (node == null)
			{
				result = null;
			}
			else
			{
				int compare = comparator.compare(value, node.value);
				if (compare == 0)
				{
					result = node.left;
				}
				else
				{
					if (compare < 0)
					{
						result = head(node.left, value, comparator);
					}
					else
					{
						result = new Node<>(node.value, node.left, head(node.right, value, comparator));
					}
				}
				result = balance(result);
			}
			return result;
		}

		public static int height(Node node)
		{
			return node == null ? 0 : node.height;
		}

		public static <Type> Node<Type> insert(Node<Type> node, Type value, Comparator<? super Type> comparator)
		{
			Node<Type> result;
			if (node == null)
			{
				result = new Node<>(value, null, null);
			}
			else
			{
				int compare = comparator.compare(value, node.value);
				if (compare == 0)
				{
					result = new Node<>(value, node.left, node.right);
					;
				}
				else
				{
					if (compare < 0)
					{
						result = new Node<>(node.value, insert(node.left, value, comparator), node.right);
					}
					else
					{
						result = new Node<>(node.value, node.left, insert(node.right, value, comparator));
					}
				}
				result = balance(result);
			}
			return result;
		}

		public static <Type> Node<Type> last(Node<Type> result)
		{
			while (result.right != null)
			{
				result = result.right;
			}
			return result;
		}

		public static int size(Node node)
		{
			return node == null ? 0 : node.size;
		}

		public static <Type> Node<Type> tail(Node<Type> node, Type value, Comparator<? super Type> comparator)
		{
			Node<Type> result;
			if (node == null)
			{
				result = null;
			}
			else
			{
				int compare = comparator.compare(value, node.value);
				if (compare == 0)
				{
					result = new Node<>(node.value, null, node.right);
				}
				else
				{
					if (compare < 0)
					{
						result = new Node<>(node.value, tail(node.left, value, comparator), node.right);
					}
					else
					{
						result = tail(node.right, value, comparator);
					}
				}
				result = balance(result);
			}
			return result;
		}

		public static <Type> void traverseOrderIn(Node<Type> node, Consumer<Type> consumer)
		{
			if (node != null)
			{
				traverseOrderIn(node.left, consumer);
				consumer.accept(node.value);
				traverseOrderIn(node.right, consumer);
			}
		}

		public final Type value;
		public final Node<Type> left;
		public final Node<Type> right;
		public final int size;
		private final int height;

		public Node(Type value, Node<Type> left, Node<Type> right)
		{
			this.value = value;
			this.left = left;
			this.right = right;
			this.size = 1 + size(left) + size(right);
			this.height = 1 + Math.max(height(left), height(right));
		}

		public Node<Type> rotateLeft()
		{
			Node<Type> left = new Node<>(this.value, this.left, this.right.left);
			return new Node<>(this.right.value, left, this.right.right);
		}

		public Node<Type> rotateRight()
		{
			Node<Type> right = new Node<>(this.value, this.left.right, this.right);
			return new Node<>(this.left.value, this.left.left, right);
		}
	}

	public static class SingleLinkedList<Type>
	{
		public final Type element;
		public SingleLinkedList<Type> next;

		public SingleLinkedList(Type element, SingleLinkedList<Type> next)
		{
			this.element = element;
			this.next = next;
		}

		public void toCollection(Collection<Type> collection)
		{
			if (this.next != null)
			{
				this.next.toCollection(collection);
			}
			collection.add(this.element);
		}
	}

	public static class SmallSetIntegers
	{
		public static final int SIZE = 20;
		public static final int[] SET = generateSet();
		public static final int[] COUNT = generateCount();
		public static final int[] INTEGER = generateInteger();

		private static int count(int set)
		{
			int result = 0;
			for (int integer = 0; integer < SIZE; integer++)
			{
				if (0 < (set & set(integer)))
				{
					result += 1;
				}
			}
			return result;
		}

		private static final int[] generateCount()
		{
			int[] result = new int[1 << SIZE];
			for (int set = 0; set < result.length; set++)
			{
				result[set] = count(set);
			}
			return result;
		}

		private static final int[] generateInteger()
		{
			int[] result = new int[1 << SIZE];
			Arrays.fill(result, -1);
			for (int integer = 0; integer < SIZE; integer++)
			{
				result[SET[integer]] = integer;
			}
			return result;
		}

		private static final int[] generateSet()
		{
			int[] result = new int[SIZE];
			for (int integer = 0; integer < result.length; integer++)
			{
				result[integer] = set(integer);
			}
			return result;
		}

		private static int set(int integer)
		{
			return 1 << integer;
		}
	}

	public static class SortedMapAVL<TypeKey, TypeValue> implements SortedMap<TypeKey, TypeValue>
	{
		public final Comparator<? super TypeKey> comparator;
		public final SortedSetAVL<Entry<TypeKey, TypeValue>> entrySet;

		public SortedMapAVL(Comparator<? super TypeKey> comparator)
		{
			this(comparator, new SortedSetAVL<>((entry0, entry1) -> comparator.compare(entry0.getKey(), entry1.getKey())));
		}

		private SortedMapAVL(Comparator<? super TypeKey> comparator, SortedSetAVL<Entry<TypeKey, TypeValue>> entrySet)
		{
			this.comparator = comparator;
			this.entrySet = entrySet;
		}

		@Override
		public void clear()
		{
			this.entrySet.clear();
		}

		@Override
		public Comparator<? super TypeKey> comparator()
		{
			return this.comparator;
		}

		@Override
		public boolean containsKey(Object key)
		{
			return this.entrySet().contains(new AbstractMap.SimpleEntry<>((TypeKey) key, null));
		}

		@Override
		public boolean containsValue(Object value)
		{
			throw new UnsupportedOperationException();
		}

		@Override
		public SortedSetAVL<Entry<TypeKey, TypeValue>> entrySet()
		{
			return this.entrySet;
		}

		public Entry<TypeKey, TypeValue> firstEntry()
		{
			return this.entrySet.first();
		}

		@Override
		public TypeKey firstKey()
		{
			return firstEntry().getKey();
		}

		@Override
		public TypeValue get(Object key)
		{
			Entry<TypeKey, TypeValue> entry = new AbstractMap.SimpleEntry<>((TypeKey) key, null);
			entry = this.entrySet.get(entry);
			return entry == null ? null : entry.getValue();
		}

		@Override
		public SortedMapAVL<TypeKey, TypeValue> headMap(TypeKey keyEnd)
		{
			return new SortedMapAVL<>(this.comparator, this.entrySet.headSet(new AbstractMap.SimpleEntry<>(keyEnd, null)));
		}

		@Override
		public boolean isEmpty()
		{
			return this.entrySet.isEmpty();
		}

		@Override
		public Set<TypeKey> keySet()
		{
			return new SortedSet<TypeKey>()
			{
				@Override
				public boolean add(TypeKey typeKey)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean addAll(Collection<? extends TypeKey> collection)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public void clear()
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public Comparator<? super TypeKey> comparator()
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean contains(Object o)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean containsAll(Collection<?> collection)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public TypeKey first()
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public SortedSet<TypeKey> headSet(TypeKey typeKey)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean isEmpty()
				{
					return size() == 0;
				}

				@Override
				public Iterator<TypeKey> iterator()
				{
					final Iterator<Entry<TypeKey, TypeValue>> iterator = SortedMapAVL.this.entrySet.iterator();
					return new Iterator<TypeKey>()
					{
						@Override
						public boolean hasNext()
						{
							return iterator.hasNext();
						}

						@Override
						public TypeKey next()
						{
							return iterator.next().getKey();
						}
					};
				}

				@Override
				public TypeKey last()
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean remove(Object o)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean removeAll(Collection<?> collection)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean retainAll(Collection<?> collection)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public int size()
				{
					return SortedMapAVL.this.entrySet.size();
				}

				@Override
				public SortedSet<TypeKey> subSet(TypeKey typeKey, TypeKey e1)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public SortedSet<TypeKey> tailSet(TypeKey typeKey)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public Object[] toArray()
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public <T> T[] toArray(T[] ts)
				{
					throw new UnsupportedOperationException();
				}
			};
		}

		public Entry<TypeKey, TypeValue> lastEntry()
		{
			return this.entrySet.last();
		}

		@Override
		public TypeKey lastKey()
		{
			return lastEntry().getKey();
		}

		@Override
		public TypeValue put(TypeKey key, TypeValue value)
		{
			TypeValue result = get(key);
			Entry<TypeKey, TypeValue> entry = new AbstractMap.SimpleEntry<>(key, value);
			this.entrySet().add(entry);
			return result;
		}

		@Override
		public void putAll(Map<? extends TypeKey, ? extends TypeValue> map)
		{
			map.entrySet()
					.forEach(entry -> put(entry.getKey(), entry.getValue()));
		}

		@Override
		public TypeValue remove(Object key)
		{
			TypeValue result = get(key);
			Entry<TypeKey, TypeValue> entry = new AbstractMap.SimpleEntry<>((TypeKey) key, null);
			this.entrySet.remove(entry);
			return result;
		}

		@Override
		public int size()
		{
			return this.entrySet().size();
		}

		@Override
		public SortedMapAVL<TypeKey, TypeValue> subMap(TypeKey keyStart, TypeKey keyEnd)
		{
			return new SortedMapAVL<>(this.comparator, this.entrySet.subSet(new AbstractMap.SimpleEntry<>(keyStart, null), new AbstractMap.SimpleEntry<>(keyEnd, null)));
		}

		@Override
		public SortedMapAVL<TypeKey, TypeValue> tailMap(TypeKey keyStart)
		{
			return new SortedMapAVL<>(this.comparator, this.entrySet.tailSet(new AbstractMap.SimpleEntry<>(keyStart, null)));
		}

		@Override
		public String toString()
		{
			return this.entrySet().toString();
		}

		@Override
		public Collection<TypeValue> values()
		{
			return new Collection<TypeValue>()
			{
				@Override
				public boolean add(TypeValue typeValue)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean addAll(Collection<? extends TypeValue> collection)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public void clear()
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean contains(Object value)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean containsAll(Collection<?> collection)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean isEmpty()
				{
					return SortedMapAVL.this.entrySet.isEmpty();
				}

				@Override
				public Iterator<TypeValue> iterator()
				{
					return new Iterator<TypeValue>()
					{
						Iterator<Entry<TypeKey, TypeValue>> iterator = SortedMapAVL.this.entrySet.iterator();

						@Override
						public boolean hasNext()
						{
							return this.iterator.hasNext();
						}

						@Override
						public TypeValue next()
						{
							return this.iterator.next().getValue();
						}
					};
				}

				@Override
				public boolean remove(Object o)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean removeAll(Collection<?> collection)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public boolean retainAll(Collection<?> collection)
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public int size()
				{
					return SortedMapAVL.this.entrySet.size();
				}

				@Override
				public Object[] toArray()
				{
					throw new UnsupportedOperationException();
				}

				@Override
				public <T> T[] toArray(T[] ts)
				{
					throw new UnsupportedOperationException();
				}
			};
		}
	}

	public static class SortedSetAVL<Type> implements SortedSet<Type>
	{
		public Comparator<? super Type> comparator;
		public Node<Type> root;

		private SortedSetAVL(Comparator<? super Type> comparator, Node<Type> root)
		{
			this.comparator = comparator;
			this.root = root;
		}

		public SortedSetAVL(Comparator<? super Type> comparator)
		{
			this(comparator, null);
		}

		public SortedSetAVL(Collection<? extends Type> collection, Comparator<? super Type> comparator)
		{
			this(comparator, null);
			this.addAll(collection);
		}

		public SortedSetAVL(SortedSetAVL<Type> sortedSetAVL)
		{
			this(sortedSetAVL.comparator, Node.clone(sortedSetAVL.root));
		}

		@Override
		public boolean add(Type value)
		{
			int sizeBefore = size();
			this.root = Node.insert(this.root, value, this.comparator);
			return sizeBefore != size();
		}

		@Override
		public boolean addAll(Collection<? extends Type> collection)
		{
			return collection.stream()
					.map(this::add)
					.reduce(true, (x, y) -> x | y);
		}

		@Override
		public void clear()
		{
			this.root = null;
		}

		@Override
		public Comparator<? super Type> comparator()
		{
			return this.comparator;
		}

		@Override
		public boolean contains(Object value)
		{
			return Node.get(this.root, (Type) value, this.comparator) != null;
		}

		@Override
		public boolean containsAll(Collection<?> collection)
		{
			return collection.stream()
					.allMatch(this::contains);
		}

		@Override
		public Type first()
		{
			return Node.first(this.root).value;
		}

		public Type get(Type value)
		{
			Node<Type> node = Node.get(this.root, value, this.comparator);
			return node == null ? null : node.value;
		}

		@Override
		public SortedSetAVL<Type> headSet(Type valueEnd)
		{
			return new SortedSetAVL<>(this.comparator, Node.head(this.root, valueEnd, this.comparator));
		}

		@Override
		public boolean isEmpty()
		{
			return this.root == null;
		}

		@Override
		public Iterator<Type> iterator()
		{
			Stack<Node<Type>> path = new Stack<>();
			return new Iterator<Type>()
			{
				{
					push(SortedSetAVL.this.root);
				}

				@Override
				public boolean hasNext()
				{
					return !path.isEmpty();
				}

				@Override
				public Type next()
				{
					if (path.isEmpty())
					{
						throw new NoSuchElementException();
					}
					else
					{
						Node<Type> node = path.peek();
						Type result = node.value;
						if (node.right != null)
						{
							push(node.right);
						}
						else
						{
							do
							{
								node = path.pop();
							}
							while (!path.isEmpty() && path.peek().right == node);
						}
						return result;
					}
				}

				public void push(Node<Type> node)
				{
					while (node != null)
					{
						path.push(node);
						node = node.left;
					}
				}
			};
		}

		@Override
		public Type last()
		{
			return Node.last(this.root).value;
		}

		@Override
		public boolean remove(Object value)
		{
			int sizeBefore = size();
			this.root = Node.delete(this.root, (Type) value, this.comparator);
			return sizeBefore != size();
		}

		@Override
		public boolean removeAll(Collection<?> collection)
		{
			return collection.stream()
					.map(this::remove)
					.reduce(true, (x, y) -> x | y);
		}

		@Override
		public boolean retainAll(Collection<?> collection)
		{
			SortedSetAVL<Type> set = new SortedSetAVL<>(this.comparator);
			collection.stream()
					.map(element -> (Type) element)
					.filter(this::contains)
					.forEach(set::add);
			boolean result = size() != set.size();
			this.root = set.root;
			return result;
		}

		@Override
		public int size()
		{
			return this.root == null ? 0 : this.root.size;
		}

		@Override
		public SortedSetAVL<Type> subSet(Type valueStart, Type valueEnd)
		{
			return tailSet(valueStart).headSet(valueEnd);
		}

		@Override
		public SortedSetAVL<Type> tailSet(Type valueStart)
		{
			return new SortedSetAVL<>(this.comparator, Node.tail(this.root, valueStart, this.comparator));
		}

		@Override
		public Object[] toArray()
		{
			return toArray(new Object[0]);
		}

		@Override
		public <T> T[] toArray(T[] ts)
		{
			List<Object> list = new ArrayList<>();
			Node.traverseOrderIn(this.root, list::add);
			return list.toArray(ts);
		}

		@Override
		public String toString()
		{
			return "{" + C_467.toString(this, ", ") + "}";
		}
	}

	public static class Tree2D
	{
		public static final int SIZE = 1 << 30;
		public static final Tree2D[] TREES_NULL = new Tree2D[] { null, null, null, null };

		public static boolean contains(int x, int y, int left, int bottom, int size)
		{
			return left <= x && x < left + size && bottom <= y && y < bottom + size;
		}

		public static int count(Tree2D[] trees)
		{
			int result = 0;
			for (int index = 0; index < 4; index++)
			{
				if (trees[index] != null)
				{
					result += trees[index].count;
				}
			}
			return result;
		}

		public static int count
				(
						int rectangleLeft,
						int rectangleBottom,
						int rectangleRight,
						int rectangleTop,
						Tree2D tree,
						int left,
						int bottom,
						int size
				)
		{
			int result;
			if (tree == null)
			{
				result = 0;
			}
			else
			{
				int right = left + size;
				int top = bottom + size;
				int intersectionLeft = Math.max(rectangleLeft, left);
				int intersectionBottom = Math.max(rectangleBottom, bottom);
				int intersectionRight = Math.min(rectangleRight, right);
				int intersectionTop = Math.min(rectangleTop, top);
				if (intersectionRight <= intersectionLeft || intersectionTop <= intersectionBottom)
				{
					result = 0;
				}
				else
				{
					if (intersectionLeft == left && intersectionBottom == bottom && intersectionRight == right && intersectionTop == top)
					{
						result = tree.count;
					}
					else
					{
						size = size >> 1;
						result = 0;
						for (int index = 0; index < 4; index++)
						{
							result += count
									(
											rectangleLeft,
											rectangleBottom,
											rectangleRight,
											rectangleTop,
											tree.trees[index],
											quadrantLeft(left, size, index),
											quadrantBottom(bottom, size, index),
											size
									);
						}
					}
				}
			}
			return result;
		}

		public static int quadrantBottom(int bottom, int size, int index)
		{
			return bottom + (index >> 1) * size;
		}

		public static int quadrantLeft(int left, int size, int index)
		{
			return left + (index & 1) * size;
		}

		public final Tree2D[] trees;
		public final int count;

		private Tree2D(Tree2D[] trees, int count)
		{
			this.trees = trees;
			this.count = count;
		}

		public Tree2D(Tree2D[] trees)
		{
			this(trees, count(trees));
		}

		public Tree2D()
		{
			this(TREES_NULL);
		}

		public int count(int rectangleLeft, int rectangleBottom, int rectangleRight, int rectangleTop)
		{
			return count
					(
							rectangleLeft,
							rectangleBottom,
							rectangleRight,
							rectangleTop,
							this,
							0,
							0,
							SIZE
					);
		}

		public Tree2D setPoint
				(
						int x,
						int y,
						Tree2D tree,
						int left,
						int bottom,
						int size
				)
		{
			Tree2D result;
			if (contains(x, y, left, bottom, size))
			{
				if (size == 1)
				{
					result = new Tree2D(TREES_NULL, 1);
				}
				else
				{
					size = size >> 1;
					Tree2D[] trees = new Tree2D[4];
					for (int index = 0; index < 4; index++)
					{
						trees[index] = setPoint
								(
										x,
										y,
										tree == null ? null : tree.trees[index],
										quadrantLeft(left, size, index),
										quadrantBottom(bottom, size, index),
										size
								);
					}
					result = new Tree2D(trees);
				}
			}
			else
			{
				result = tree;
			}
			return result;
		}

		public Tree2D setPoint(int x, int y)
		{
			return setPoint
					(
							x,
							y,
							this,
							0,
							0,
							SIZE
					);
		}
	}

	public static class Tuple2<Type0, Type1>
	{
		public final Type0 v0;
		public final Type1 v1;

		public Tuple2(Type0 v0, Type1 v1)
		{
			this.v0 = v0;
			this.v1 = v1;
		}

		@Override
		public String toString()
		{
			return "(" + this.v0 + ", " + this.v1 + ")";
		}
	}

	public static class Tuple2Comparable<Type0 extends Comparable<? super Type0>, Type1 extends Comparable<? super Type1>> extends Tuple2<Type0, Type1> implements Comparable<Tuple2Comparable<Type0, Type1>>
	{
		public Tuple2Comparable(Type0 v0, Type1 v1)
		{
			super(v0, v1);
		}

		@Override
		public int compareTo(Tuple2Comparable<Type0, Type1> that)
		{
			int result = this.v0.compareTo(that.v0);
			if (result == 0)
			{
				result = this.v1.compareTo(that.v1);
			}
			return result;
		}
	}

	public static class Tuple3<Type0, Type1, Type2>
	{
		public final Type0 v0;
		public final Type1 v1;
		public final Type2 v2;

		public Tuple3(Type0 v0, Type1 v1, Type2 v2)
		{
			this.v0 = v0;
			this.v1 = v1;
			this.v2 = v2;
		}

		@Override
		public String toString()
		{
			return "(" + this.v0 + ", " + this.v1 + ", " + this.v2 + ")";
		}
	}

	public static class Vertex
			<
					TypeVertex extends Vertex<TypeVertex, TypeEdge>,
					TypeEdge extends Edge<TypeVertex, TypeEdge>
					>
			implements Comparable<Vertex<? super TypeVertex, ? super TypeEdge>>
	{
		public static <
				TypeVertex extends Vertex<TypeVertex, TypeEdge>,
				TypeEdge extends Edge<TypeVertex, TypeEdge>,
				TypeResult
				> TypeResult breadthFirstSearch
				(
						TypeVertex vertex,
						TypeEdge edge,
						BiFunctionResult<TypeVertex, TypeEdge, TypeResult> function,
						Array<Boolean> visited,
						FIFO<TypeVertex> verticesNext,
						FIFO<TypeEdge> edgesNext,
						TypeResult result
				)
		{
			if (!visited.get(vertex.index))
			{
				visited.set(vertex.index, true);
				result = function.apply(vertex, edge, result);
				for (TypeEdge edgeNext : vertex.edges)
				{
					TypeVertex vertexNext = edgeNext.other(vertex);
					if (!visited.get(vertexNext.index))
					{
						verticesNext.push(vertexNext);
						edgesNext.push(edgeNext);
					}
				}
			}
			return result;
		}

		public static <
				TypeVertex extends Vertex<TypeVertex, TypeEdge>,
				TypeEdge extends Edge<TypeVertex, TypeEdge>,
				TypeResult
				>
		TypeResult breadthFirstSearch
				(
						Array<TypeVertex> vertices,
						int indexVertexStart,
						BiFunctionResult<TypeVertex, TypeEdge, TypeResult> function,
						TypeResult result
				)
		{
			Array<Boolean> visited = new Array<>(vertices.size(), false);
			FIFO<TypeVertex> verticesNext = new FIFO<>();
			verticesNext.push(vertices.get(indexVertexStart));
			FIFO<TypeEdge> edgesNext = new FIFO<>();
			edgesNext.push(null);
			while (!verticesNext.isEmpty())
			{
				result = breadthFirstSearch(verticesNext.pop(), edgesNext.pop(), function, visited, verticesNext, edgesNext, result);
			}
			return result;
		}

		public static <
				TypeVertex extends Vertex<TypeVertex, TypeEdge>,
				TypeEdge extends Edge<TypeVertex, TypeEdge>
				>
		boolean
		cycle
				(
						TypeVertex start,
						SortedSet<TypeVertex> result
				)
		{
			boolean cycle = false;
			Stack<TypeVertex> stackVertex = new Stack<>();
			Stack<TypeEdge> stackEdge = new Stack<>();
			stackVertex.push(start);
			stackEdge.push(null);
			while (!stackVertex.isEmpty())
			{
				TypeVertex vertex = stackVertex.pop();
				TypeEdge edge = stackEdge.pop();
				if (!result.contains(vertex))
				{
					result.add(vertex);
					for (TypeEdge otherEdge : vertex.edges)
					{
						if (otherEdge != edge)
						{
							TypeVertex otherVertex = otherEdge.other(vertex);
							if (result.contains(otherVertex))
							{
								cycle = true;
							}
							else
							{
								stackVertex.push(otherVertex);
								stackEdge.push(otherEdge);
							}
						}
					}
				}
			}
			return cycle;
		}

		public static <
				TypeVertex extends Vertex<TypeVertex, TypeEdge>,
				TypeEdge extends Edge<TypeVertex, TypeEdge>
				>
		SortedSet<TypeVertex>
		depthFirstSearch
				(
						TypeVertex start,
						BiConsumer<TypeVertex, TypeEdge> functionVisitPre,
						BiConsumer<TypeVertex, TypeEdge> functionVisitPost
				)
		{
			SortedSet<TypeVertex> result = new SortedSetAVL<>(Comparator.naturalOrder());
			Stack<TypeVertex> stackVertex = new Stack<>();
			Stack<TypeEdge> stackEdge = new Stack<>();
			stackVertex.push(start);
			stackEdge.push(null);
			while (!stackVertex.isEmpty())
			{
				TypeVertex vertex = stackVertex.pop();
				TypeEdge edge = stackEdge.pop();
				if (result.contains(vertex))
				{
					functionVisitPost.accept(vertex, edge);
				}
				else
				{
					result.add(vertex);
					stackVertex.push(vertex);
					stackEdge.push(edge);
					functionVisitPre.accept(vertex, edge);
					for (TypeEdge otherEdge : vertex.edges)
					{
						TypeVertex otherVertex = otherEdge.other(vertex);
						if (!result.contains(otherVertex))
						{
							stackVertex.push(otherVertex);
							stackEdge.push(otherEdge);
						}
					}
				}
			}
			return result;
		}

		public static <
				TypeVertex extends Vertex<TypeVertex, TypeEdge>,
				TypeEdge extends Edge<TypeVertex, TypeEdge>
				>
		SortedSet<TypeVertex>
		depthFirstSearch
				(
						TypeVertex start,
						Consumer<TypeVertex> functionVisitPreVertex,
						Consumer<TypeVertex> functionVisitPostVertex
				)
		{
			BiConsumer<TypeVertex, TypeEdge> functionVisitPreVertexEdge = (vertex, edge) ->
			{
				functionVisitPreVertex.accept(vertex);
			};
			BiConsumer<TypeVertex, TypeEdge> functionVisitPostVertexEdge = (vertex, edge) ->
			{
				functionVisitPostVertex.accept(vertex);
			};
			return depthFirstSearch(start, functionVisitPreVertexEdge, functionVisitPostVertexEdge);
		}

		public final int index;
		public final List<TypeEdge> edges;

		public Vertex(int index)
		{
			this.index = index;
			this.edges = new ArrayList<>();
		}

		@Override
		public int compareTo(Vertex<? super TypeVertex, ? super TypeEdge> that)
		{
			return Integer.compare(this.index, that.index);
		}

		@Override
		public String toString()
		{
			return "" + this.index;
		}
	}

	public static class VertexDefault<TypeEdge extends Edge<VertexDefault<TypeEdge>, TypeEdge>> extends Vertex<VertexDefault<TypeEdge>, TypeEdge>
	{
		public VertexDefault(int index)
		{
			super(index);
		}
	}

	public static class VertexDefaultDefault extends Vertex<VertexDefaultDefault, EdgeDefaultDefault>
	{
		public static Array<VertexDefaultDefault> vertices(int n)
		{
			Array<VertexDefaultDefault> result = new Array<>(n);
			for (int index = 0; index < n; index++)
			{
				result.set(index, new VertexDefaultDefault(index));
			}
			return result;
		}

		public VertexDefaultDefault(int index)
		{
			super(index);
		}
	}

	public static class Wrapper<Type>
	{
		public Type value;

		public Wrapper(Type value)
		{
			this.value = value;
		}

		public Type get()
		{
			return this.value;
		}

		public void set(Type value)
		{
			this.value = value;
		}

		@Override
		public String toString()
		{
			return this.value.toString();
		}
	}

	public static void add(int delta, int[] result)
	{
		for (int index = 0; index < result.length; index++)
		{
			result[index] += delta;
		}
	}

	public static void add(int delta, int[]... result)
	{
		for (int index = 0; index < result.length; index++)
		{
			add(delta, result[index]);
		}
	}

	public static long add(long x, long y)
	{
		return (x + y) % BIG;
	}

	public static int binarySearchMaximum(Function<Integer, Boolean> filter, int start, int end)
	{
		return -binarySearchMinimum(x -> filter.apply(-x), -end, -start);
	}

	public static int binarySearchMinimum(Function<Integer, Boolean> filter, int start, int end)
	{
		int result;
		if (start == end)
		{
			result = end;
		}
		else
		{
			int middle = start + (end - start) / 2;
			if (filter.apply(middle))
			{
				result = binarySearchMinimum(filter, start, middle);
			}
			else
			{
				result = binarySearchMinimum(filter, middle + 1, end);
			}
		}
		return result;
	}

	public static void close()
	{
		out.close();
	}

	private static void combinations(int n, int k, int start, SortedSet<Integer> combination, List<SortedSet<Integer>> result)
	{
		if (k == 0)
		{
			result.add(new SortedSetAVL<>(combination, Comparator.naturalOrder()));
		}
		else
		{
			for (int index = start; index < n; index++)
			{
				if (!combination.contains(index))
				{
					combination.add(index);
					combinations(n, k - 1, index + 1, combination, result);
					combination.remove(index);
				}
			}
		}
	}

	public static List<SortedSet<Integer>> combinations(int n, int k)
	{
		List<SortedSet<Integer>> result = new ArrayList<>();
		combinations(n, k, 0, new SortedSetAVL<>(Comparator.naturalOrder()), result);
		return result;
	}

	public static <Type> int compare(Iterator<Type> iterator0, Iterator<Type> iterator1, Comparator<Type> comparator)
	{
		int result = 0;
		while (result == 0 && iterator0.hasNext() && iterator1.hasNext())
		{
			result = comparator.compare(iterator0.next(), iterator1.next());
		}
		if (result == 0)
		{
			if (iterator1.hasNext())
			{
				result = -1;
			}
			else
			{
				if (iterator0.hasNext())
				{
					result = 1;
				}
			}
		}
		return result;
	}

	public static <Type> int compare(Iterable<Type> iterable0, Iterable<Type> iterable1, Comparator<Type> comparator)
	{
		return compare(iterable0.iterator(), iterable1.iterator(), comparator);
	}

	public static long divideCeil(long x, long y)
	{
		return (x + y - 1) / y;
	}

	public static Set<Long> divisors(long n)
	{
		SortedSetAVL<Long> result = new SortedSetAVL<>(Comparator.naturalOrder());
		result.add(1L);
		for (Long factor : factors(n))
		{
			SortedSetAVL<Long> divisors = new SortedSetAVL<>(result);
			for (Long divisor : result)
			{
				divisors.add(divisor * factor);
			}
			result = divisors;
		}
		return result;
	}

	public static LinkedList<Long> factors(long n)
	{
		LinkedList<Long> result = new LinkedList<>();
		Iterator<Long> primes = ITERATOR_BUFFER_PRIME.iterator();
		Long prime;
		while (n > 1 && (prime = primes.next()) * prime <= n)
		{
			while (n % prime == 0)
			{
				result.add(prime);
				n /= prime;
			}
		}
		if (n > 1)
		{
			result.add(n);
		}
		return result;
	}

	public static long faculty(int n)
	{
		long result = 1;
		for (int index = 2; index <= n; index++)
		{
			result *= index;
		}
		return result;
	}

	static long gcd(long a, long b)
	{
		return b == 0 ? a : gcd(b, a % b);
	}

	public static long[] generatePOWER2()
	{
		long[] result = new long[63];
		for (int x = 0; x < result.length; x++)
		{
			result[x] = 1L << x;
		}
		return result;
	}

	public static boolean isPrime(long x)
	{
		boolean result = x > 1;
		Iterator<Long> iterator = ITERATOR_BUFFER_PRIME.iterator();
		Long prime;
		while ((prime = iterator.next()) * prime <= x)
		{
			result &= x % prime > 0;
		}
		return result;
	}

	public static long knapsack(List<Tuple3<Long, Integer, Integer>> itemsValueWeightCount, int weightMaximum)
	{
		long[] valuesMaximum = new long[weightMaximum + 1];
		for (Tuple3<Long, Integer, Integer> itemValueWeightCount : itemsValueWeightCount)
		{
			long itemValue = itemValueWeightCount.v0;
			int itemWeight = itemValueWeightCount.v1;
			int itemCount = itemValueWeightCount.v2;
			for (int weight = weightMaximum; 0 <= weight; weight--)
			{
				for (int index = 1; index <= itemCount && 0 <= weight - index * itemWeight; index++)
				{
					valuesMaximum[weight] = Math.max(valuesMaximum[weight], valuesMaximum[weight - index * itemWeight] + index * itemValue);
				}
			}
		}
		long result = 0;
		for (long valueMaximum : valuesMaximum)
		{
			result = Math.max(result, valueMaximum);
		}
		return result;
	}

	public static boolean knapsackPossible(List<Tuple2<Integer, Integer>> itemsWeightCount, int weightMaximum)
	{
		boolean[] weightPossible = new boolean[weightMaximum + 1];
		weightPossible[0] = true;
		int weightLargest = 0;
		for (Tuple2<Integer, Integer> itemWeightCount : itemsWeightCount)
		{
			int itemWeight = itemWeightCount.v0;
			int itemCount = itemWeightCount.v1;
			for (int weightStart = 0; weightStart < itemWeight; weightStart++)
			{
				int count = 0;
				for (int weight = weightStart; weight <= weightMaximum && (0 < count || weight <= weightLargest); weight += itemWeight)
				{
					if (weightPossible[weight])
					{
						count = itemCount;
					}
					else
					{
						if (0 < count)
						{
							weightPossible[weight] = true;
							weightLargest = weight;
							count -= 1;
						}
					}
				}
			}
		}
		return weightPossible[weightMaximum];
	}

	public static long lcm(int a, int b)
	{
		return a * b / gcd(a, b);
	}

	public static void main(String[] args)
	{
		try
		{
			solve();
		}
		catch (IOException exception)
		{
			exception.printStackTrace();
		}
		close();
	}

	public static long mul(long x, long y)
	{
		return (x * y) % BIG;
	}

	public static double nextDouble() throws IOException
	{
		return Double.parseDouble(nextString());
	}

	public static int nextInt() throws IOException
	{
		return Integer.parseInt(nextString());
	}

	public static void nextInts(int n, int[]... result) throws IOException
	{
		for (int index = 0; index < n; index++)
		{
			for (int value = 0; value < result.length; value++)
			{
				result[value][index] = nextInt();
			}
		}
	}

	public static int[] nextInts(int n) throws IOException
	{
		int[] result = new int[n];
		nextInts(n, result);
		return result;
	}

	public static String nextLine() throws IOException
	{
		return bufferedReader.readLine();
	}

	public static long nextLong() throws IOException
	{
		return Long.parseLong(nextString());
	}

	public static void nextLongs(int n, long[]... result) throws IOException
	{
		for (int index = 0; index < n; index++)
		{
			for (int value = 0; value < result.length; value++)
			{
				result[value][index] = nextLong();
			}
		}
	}

	public static long[] nextLongs(int n) throws IOException
	{
		long[] result = new long[n];
		nextLongs(n, result);
		return result;
	}

	public static String nextString() throws IOException
	{
		while ((stringTokenizer == null) || (!stringTokenizer.hasMoreTokens()))
		{
			stringTokenizer = new StringTokenizer(bufferedReader.readLine());
		}
		return stringTokenizer.nextToken();
	}

	public static String[] nextStrings(int n) throws IOException
	{
		String[] result = new String[n];
		{
			for (int index = 0; index < n; index++)
			{
				result[index] = nextString();
			}
		}
		return result;
	}

	public static <T> List<T> permutation(long p, List<T> x)
	{
		List<T> copy = new ArrayList<>();
		for (int index = 0; index < x.size(); index++)
		{
			copy.add(x.get(index));
		}
		List<T> result = new ArrayList<>();
		for (int indexTo = 0; indexTo < x.size(); indexTo++)
		{
			int indexFrom = (int) p % copy.size();
			p = p / copy.size();
			result.add(copy.remove(indexFrom));
		}
		return result;
	}

	public static <Type> List<List<Type>> permutations(List<Type> list)
	{
		List<List<Type>> result = new ArrayList<>();
		result.add(new ArrayList<>());
		for (Type element : list)
		{
			List<List<Type>> permutations = result;
			result = new ArrayList<>();
			for (List<Type> permutation : permutations)
			{
				for (int index = 0; index <= permutation.size(); index++)
				{
					List<Type> permutationNew = new ArrayList<>(permutation);
					permutationNew.add(index, element);
					result.add(permutationNew);
				}
			}
		}
		return result;
	}

	public static long[][] sizeGroup2CombinationsCount(int maximum)
	{
		long[][] result = new long[maximum + 1][maximum + 1];
		for (int length = 0; length <= maximum; length++)
		{
			result[length][0] = 1;
			for (int group = 1; group <= length; group++)
			{
				result[length][group] = add(result[length - 1][group - 1], result[length - 1][group]);
			}
		}
		return result;
	}

	public static Stream<BigInteger> streamFibonacci()
	{
		return Stream.generate(new Supplier<BigInteger>()
		{
			private BigInteger n0 = BigInteger.ZERO;
			private BigInteger n1 = BigInteger.ONE;

			@Override
			public BigInteger get()
			{
				BigInteger result = n0;
				n0 = n1;
				n1 = result.add(n0);
				return result;
			}
		});
	}

	public static Stream<Long> streamPrime(int sieveSize)
	{
		return Stream.generate(new Supplier<Long>()
		{
			private boolean[] isPrime = new boolean[sieveSize];
			private long sieveOffset = 2;
			private List<Long> primes = new ArrayList<>();
			private int index = 0;

			public void filter(long prime, boolean[] result)
			{
				if (prime * prime < this.sieveOffset + sieveSize)
				{
					long remainingStart = this.sieveOffset % prime;
					long start = remainingStart == 0 ? 0 : prime - remainingStart;
					for (long index = start; index < sieveSize; index += prime)
					{
						result[(int) index] = false;
					}
				}
			}

			public void generatePrimes()
			{
				Arrays.fill(this.isPrime, true);
				this.primes.forEach(prime -> filter(prime, isPrime));
				for (int index = 0; index < sieveSize; index++)
				{
					if (isPrime[index])
					{
						this.primes.add(this.sieveOffset + index);
						filter(this.sieveOffset + index, isPrime);
					}
				}
				this.sieveOffset += sieveSize;
			}

			@Override
			public Long get()
			{
				while (this.primes.size() <= this.index)
				{
					generatePrimes();
				}
				Long result = this.primes.get(this.index);
				this.index += 1;
				return result;
			}
		});
	}

	public static <Type> String toString(Iterator<Type> iterator, String separator)
	{
		StringBuilder stringBuilder = new StringBuilder();
		if (iterator.hasNext())
		{
			stringBuilder.append(iterator.next());
		}
		while (iterator.hasNext())
		{
			stringBuilder.append(separator);
			stringBuilder.append(iterator.next());
		}
		return stringBuilder.toString();
	}

	public static <Type> String toString(Iterator<Type> iterator)
	{
		return toString(iterator, " ");
	}

	public static <Type> String toString(Iterable<Type> iterable, String separator)
	{
		return toString(iterable.iterator(), separator);
	}

	public static <Type> String toString(Iterable<Type> iterable)
	{
		return toString(iterable, " ");
	}

	public static long totient(long n)
	{
		Set<Long> factors = new SortedSetAVL<>(factors(n), Comparator.naturalOrder());
		long result = n;
		for (long p : factors)
		{
			result -= result / p;
		}
		return result;
	}

	interface BiFunctionResult<Type0, Type1, TypeResult>
	{
		TypeResult apply(Type0 x0, Type1 x1, TypeResult x2);
	}

	public static long periodStove(long periodStoveOn, long periodJuliaChecks)
	{
		long result;
		long juliaFindsStoveOn = periodStoveOn / periodJuliaChecks;
		if (periodStoveOn % periodJuliaChecks == 0)
		{
			result = periodStoveOn;
		}
		else
		{
			result = (juliaFindsStoveOn + 1) * periodJuliaChecks;
		}
		return result;
	}

	public static void solve() throws IOException
	{
		double heatStoveOn = 2;
		double heatStoveOff = 1;
		long periodStoveOn = nextLong();
		long periodJuliaChecks = nextLong();
		double heatChickenNeeds = heatStoveOn * nextLong();
		double result;
		long periodStove = periodStove(periodStoveOn, periodJuliaChecks);
		double heatPerPeriod = heatStoveOn * periodStoveOn + heatStoveOff * (periodStove - periodStoveOn);
		double fullPeriods = Math.floor(heatChickenNeeds / heatPerPeriod);
		result = fullPeriods * periodStove;
		heatChickenNeeds -= fullPeriods * heatPerPeriod;
		double extraOn = Math.min(periodStoveOn, heatChickenNeeds / heatStoveOn);
		result += extraOn;
		heatChickenNeeds -= extraOn * heatStoveOn;
		double extraOff = heatChickenNeeds / heatStoveOff;
		result += extraOff;
		out.println(result);
	}
}


import java.util.Scanner;

public class C {

	static double del = 1e-9;
	public static void main(String[] args) {
		Scanner in = new Scanner(System.in);

		long k = in.nextLong();
		long d = in.nextLong();
		long t = in.nextLong();

		long on = k;
		long off = (k%d==0? 0: d-k%d);
		long time = on+off;
		double perCyc = on+.5*off;
		long cyc = (long) (t/perCyc);
		double burn = cyc*perCyc;
		double tt = cyc*time;

		if(burn >= t-del){
			System.out.println(tt);
		}
		else{
			if(burn + on >= t-del){
				double newT = tt+(t-burn);
				System.out.println(newT);
			}
			else{
				burn += on;
				tt+=on;
				double newT = tt + (t-burn)*2;
				System.out.println(newT);

			}

		}

		//System.out.println(b+" "+time);

	}
	static boolean eq(double a, double b){
		if(Math.abs(a-b) < 1e-9) return true;
		return false;
	}
}


import java.util.Scanner;

public class C {

	static double del = 1e-9;
	public static void main(String[] args) {
		Scanner in = new Scanner(System.in);

		long k = in.nextLong();
		long d = in.nextLong();
		long t = in.nextLong();

		long on = k;
		long off = (k%d==0? 0: d-k%d);
		long time = on+off;
		double perCyc = on+.5*off;
		long cyc = (long) (t/perCyc);
		long b = time*cyc;
		double curCook = cyc*perCyc;

		if(curCook >= t-del){
			System.out.println(curCook);
		}
		else{
		//	System.out.println("1: "+curCook);
			curCook += on;
		//	System.out.println("2: "+curCook);
			if(curCook >= t-del){
				System.out.println(curCook);
			}
			else{
				curCook += off*.5;
				System.out.println(curCook);
			}

		}
		//System.out.println(b+" "+time);

	}
	static boolean eq(double a, double b){
		if(Math.abs(a-b) < 1e-9) return true;
		return false;
	}
}


import java.math.BigInteger;
import java.util.Scanner;

public class c {
	public static void main(String[] args) {
		Scanner scan = new Scanner(System.in);
		long k = scan.nextLong();
		long d = scan.nextLong();
		long t = scan.nextLong();
		BigInteger add = BigInteger.ZERO;
		if(k % d != 0)
			add = BigInteger.ONE;

		BigInteger kk = new BigInteger("" + k);
		BigInteger dd = new BigInteger("" + d);
		BigInteger tt = new BigInteger("" + t);
		tt = tt.multiply(new BigInteger("2"));

		BigInteger D = dd.multiply( (kk.divide(dd)).add(add) );
		BigInteger P = kk.add(D); // this is *2

//		System.out.println(D + " " + P);

		BigInteger res = tt.divide(P);
		res = res.multiply(P);

		// at the beginning of a pattern

		tt = tt.subtract(res);

		long lo = 0, hi = (long)1e18;
		for(int bs = 0; bs < 100; bs++) {
			long mid = (lo+hi)>>1;
			long time = Math.min(k, mid)*2+Math.max(mid-k, 0);
			if(tt.longValue() > time)
				lo = mid;
			else
				hi = mid;
		}
		long fuck = Math.min(k, hi)*2+Math.max(hi-k, 0);
//		System.out.println(res+" "+fuck);
		System.out.println((double)(res.longValue()+fuck)/2.0);
	}
}


import java.math.BigDecimal;
import java.math.BigInteger;
import java.math.RoundingMode;
import java.util.Scanner;

public class c {
	public static void main(String[] args) {
		Scanner scan = new Scanner(System.in);
		long k = scan.nextLong();
		long d = scan.nextLong();
		long t = scan.nextLong();
		BigInteger add = BigInteger.ZERO;
		if(k % d != 0)
			add = BigInteger.ONE;

		BigInteger kk = new BigInteger("" + k);
		BigInteger dd = new BigInteger("" + d);
		BigInteger tt = new BigInteger("" + t);
		tt = tt.multiply(new BigInteger("2"));

		BigInteger D = dd.multiply( (kk.divide(dd)).add(add) );
		BigInteger P = kk.add(D); // this is *2

//		System.out.println(D + " " + P);

		BigInteger res = tt.divide(P);
		res = res.multiply(P);

		// at the beginning of a pattern

		tt = tt.subtract(res);

		BigInteger lo = BigInteger.ZERO, hi = new BigInteger("1000000000000000000");
		for(int bs = 0; bs < 200; bs++) {
			BigInteger mid = (lo.add(hi)).divide(new BigInteger("2"));
			BigInteger time = new BigInteger("0");
			if(kk.compareTo(mid) < 0)
				time = time.add((kk).multiply(new BigInteger("2")));
			else
				time = time.add(mid.multiply(new BigInteger("2")));

			if((mid.subtract(kk)).compareTo(BigInteger.ZERO) > 0)
				time = time.add(mid.subtract(kk));

			if(tt.compareTo(time) > 0)
				lo = mid;
			else
				hi = mid;
		}
		BigInteger fuck = BigInteger.ZERO;
		if(kk.compareTo(hi) < 0)
			fuck = fuck.add((kk).multiply(new BigInteger("2")));
		else
			fuck = fuck.add(hi.multiply(new BigInteger("2")));

		add = BigInteger.ZERO;
		if((hi.subtract(kk)).compareTo(BigInteger.ZERO) > 0)
			add = hi.subtract(kk);
//		System.out.println(res+" "+fuck);
		BigDecimal fuckfuck = new BigDecimal(fuck.add(res));
		fuckfuck = fuckfuck.divide(new BigDecimal("2"), 5, RoundingMode.HALF_UP);
		fuckfuck = fuckfuck.add(new BigDecimal(add));
		System.out.println(fuckfuck);
	}
}


import java.math.BigInteger;
import java.util.Scanner;

public class c {
	public static void main(String[] args) {
		Scanner scan = new Scanner(System.in);
		long k = scan.nextLong();
		long d = scan.nextLong();
		long t = scan.nextLong();
		BigInteger add = BigInteger.ZERO;
		if(k % d != 0)
			add = BigInteger.ONE;

		BigInteger kk = new BigInteger("" + k);
		BigInteger dd = new BigInteger("" + d);
		BigInteger tt = new BigInteger("" + t);
		tt = tt.multiply(new BigInteger("2"));

		BigInteger D = dd.multiply( (kk.divide(dd)).add(add) );
		BigInteger P = kk.add(D); // this is *2

//		System.out.println(D + " " + P);

		BigInteger res = tt.divide(P);
		res = res.multiply(P);

		// at the beginning of a pattern

		tt = tt.subtract(res);

		BigInteger lo = BigInteger.ZERO, hi = new BigInteger("1000000000000000000");
		for(int bs = 0; bs < 200; bs++) {
			BigInteger mid = (lo.add(hi)).divide(new BigInteger("2"));
			BigInteger time = new BigInteger("0");
			if(kk.compareTo(mid) < 0)
				time = time.add((kk).multiply(new BigInteger("2")));
			else
				time = time.add(mid.multiply(new BigInteger("2")));

			if((mid.subtract(kk)).compareTo(BigInteger.ZERO) > 0)
				time = time.add(mid.subtract(kk));

			if(tt.compareTo(time) > 0)
				lo = mid;
			else
				hi = mid;
		}
		BigInteger fuck = BigInteger.ZERO;
		if(kk.compareTo(hi) < 0)
			fuck = fuck.add((kk).multiply(new BigInteger("2")));
		else
			fuck = fuck.add(hi.multiply(new BigInteger("2")));

		add = BigInteger.ZERO;
		if((hi.subtract(kk)).compareTo(BigInteger.ZERO) > 0)
			add = hi.subtract(kk);
//		System.out.println(res+" "+fuck);
		System.out.println(((fuck.add(res)).doubleValue() / 2.0) + add.doubleValue());
	}
}


import java.math.BigInteger;
import java.util.Scanner;

public class c {
	public static void main(String[] args) {
		Scanner scan = new Scanner(System.in);
		long k = scan.nextLong();
		long d = scan.nextLong();
		long t = scan.nextLong();
		BigInteger add = BigInteger.ZERO;
		if(k % d != 0)
			add = BigInteger.ONE;

		BigInteger kk = new BigInteger("" + k);
		BigInteger dd = new BigInteger("" + d);
		BigInteger tt = new BigInteger("" + t);
		tt = tt.multiply(new BigInteger("2"));

		BigInteger D = dd.multiply( (kk.divide(dd)).add(add) );
		BigInteger P = kk.add(D); // this is *2

//		System.out.println(D + " " + P);

		BigInteger res = tt.divide(P);
		res = res.multiply(P);

		// at the beginning of a pattern

		tt = tt.subtract(res);

		BigInteger lo = BigInteger.ZERO, hi = new BigInteger("1000000000000000000");
		for(int bs = 0; bs < 200; bs++) {
			BigInteger mid = (lo.add(hi)).divide(new BigInteger("2"));
			BigInteger time = new BigInteger("0");
			if(kk.compareTo(mid) < 0)
				time = time.add((kk).multiply(new BigInteger("2")));
			else
				time = time.add(mid.multiply(new BigInteger("2")));

			if((mid.subtract(kk)).compareTo(BigInteger.ZERO) > 0)
				time = time.add(mid.subtract(kk));

			if(tt.compareTo(time) > 0)
				lo = mid;
			else
				hi = mid;
		}
		BigInteger fuck = BigInteger.ZERO;
		if(kk.compareTo(hi) < 0)
			fuck = fuck.add((kk).multiply(new BigInteger("2")));
		else
			fuck = fuck.add(hi.multiply(new BigInteger("2")));

		if((hi.subtract(kk)).compareTo(BigInteger.ZERO) > 0)
			fuck = fuck.add(hi.subtract(kk));
//		System.out.println(res+" "+fuck);
		System.out.println((fuck.add(res)).doubleValue() / 2.0);
	}
}


import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.InputMismatchException;
import java.io.IOException;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 *
 * @author ankur
 */
public class Main {
    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        InputReader in = new InputReader(inputStream);
        PrintWriter out = new PrintWriter(outputStream);
        TaskC solver = new TaskC();
        solver.solve(1, in, out);
        out.close();
    }

    static class TaskC {
        public void solve(int testNumber, InputReader in, PrintWriter out) {
            double k = in.nextDouble();
            double d = in.nextDouble();
            long t = in.nextLong();
            double gap = Math.ceil(k / d);
            gap = gap * d;
            gap = gap - k;
            double proper = t;
            double num = k + gap / 2;
            double intcycle = (long) (proper / num);
            double ans = intcycle * (k + gap);
            if (Math.abs(intcycle * (num + gap / 2) - t) < 1e-10) {
                out.printf("%.10f", ans);
                return;
            }
            double needed = proper - intcycle * (num);
            if (needed > k) {
                needed = needed - k;
                // double ans=intcycle;
                ans += k;
                ans += needed * 2;
                out.printf("%.10f", ans);
                return;
            } else {
                out.printf("%.10f", (ans + needed));
                return;
            }
        }

    }

    static class InputReader {
        private final InputStream stream;
        private final byte[] buf = new byte[8192];
        private int curChar;
        private int snumChars;

        public InputReader(InputStream st) {
            this.stream = st;
        }

        public int read() {
            //*-*------clare------
//remeber while comparing 2 non primitive data type not used ==
//remember Arrays.sort for primitive data has worst time case complexity of 0(n^2) bcoz it uses quick sort
//again silly mistakes ,yr kb tk krta rhega ye mistakes
//try to write simple codes ,break it into simple things
//knowledge>rating
        /*
        public class Main
        implements Runnable{
    public static void main(String[] args) {
        new Thread(null,new Main(),"Main",1<<26).start();

    }
    public void run() {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        InputReader in = new InputReader(inputStream);
        PrintWriter out = new PrintWriter(outputStream);
        TaskC solver = new TaskC();//chenge the name of task
        solver.solve(1, in, out);
        out.close();
    }
         */

            if (snumChars == -1)
                throw new InputMismatchException();
            if (curChar >= snumChars) {
                curChar = 0;
                try {
                    snumChars = stream.read(buf);
                } catch (IOException e) {
                    throw new InputMismatchException();
                }
                if (snumChars <= 0)
                    return -1;
            }
            return buf[curChar++];
        }

        public double nextDouble() {
            String doub = readString();
            return Double.parseDouble(doub);
        }

        public long nextLong() {
            int c = read();
            while (isSpaceChar(c)) {
                c = read();
            }
            int sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = read();
            }
            long res = 0;
            do {
                res *= 10;
                res += c - '0';
                c = read();
            } while (!isSpaceChar(c));
            return res * sgn;
        }

        public String readString() {
            int c = read();
            while (isSpaceChar(c)) {
                c = read();
            }
            StringBuilder res = new StringBuilder();
            do {
                res.appendCodePoint(c);
                c = read();
            } while (!isSpaceChar(c));
            return res.toString();
        }

        public boolean isSpaceChar(int c) {
            return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
        }

    }
}


import java.util.Scanner;

public class C {

    private static int k, d, t;

    private static double calc(int periodOn, int periodOff) {
        int periodT = periodOn + periodOff;
        int periodCook = periodOn*2 + periodOff;

        int remainderT = t - (t/periodCook) * periodCook;
        if (remainderT <= 2*periodOn)
            return (t/periodCook)*periodT + (remainderT/2.);
        else
            return (t/periodCook)*periodT + periodOn + (remainderT - periodOn*2);
    }

    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);

        k = in.nextInt();
        d = in.nextInt();
        t = in.nextInt() * 2;

        if (k <= d)
            System.out.println(calc(d, d - k));
        else
            System.out.println(calc(k, ((k + d - 1)/d)*d - k));
    }

}


package cr467div2;

import java.util.Scanner;

public class C {

    private static long k, d, t;

    private static double calc(long periodOn, long periodOff) {
        long periodT = periodOn + periodOff;
        long periodCook = periodOn*2 + periodOff;

        long remainderT = t - floorDiv(t, periodCook)*periodCook;
        if (remainderT <= 2*periodOn)
            return floorDiv(t, periodCook)*periodT + (remainderT/2.);
        else
            return floorDiv(t, periodCook)*periodT + periodOn + (remainderT - periodOn*2);
    }

    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);

        k = in.nextLong();
        d = in.nextLong();
        t = in.nextLong() * 2;

        if (k <= d)
            System.out.println(calc(k, d - k));
        else
            System.out.println(calc(k, ceilDiv(k, d)*d - k));
    }

    private static long floorDiv(long a, long b) {
        return a / b;
    }

    private static long ceilDiv(long a, long b) {
        return (a + b - 1) / b;
    }

}


import java.util.Scanner;

public class C {

    private static long k, d, t;

    private static double calc(long periodOn, long periodOff) {
        long periodT = periodOn + periodOff;
        long periodCook = periodOn*2 + periodOff;

        long remainderT = t - floorDiv(t, periodCook)*periodCook;
        if (remainderT <= 2*periodOn)
            return floorDiv(t, periodCook)*periodT + (remainderT/2.);
        else
            return floorDiv(t, periodCook)*periodT + periodOn + (remainderT - periodOn*2);
    }

    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);

        k = in.nextLong();
        d = in.nextLong();
        t = in.nextLong() * 2;

        if (k <= d)
            System.out.println(calc(k, d - k));
        else
            System.out.println(calc(k, ceilDiv(k, d)*d - k));
    }

    private static long floorDiv(long a, long b) {
        return a / b;
    }

    private static long ceilDiv(long a, long b) {
        return (a + b - 1) / b;
    }

}


import java.util.Scanner;

public class C {

    private static int k, d, t;

    private static double calc(int periodOn, int periodOff) {
        int periodT = periodOn + periodOff;
        int periodCook = periodOn*2 + periodOff;

        int remainderT = t - (t/periodCook) * periodCook;
        if (remainderT <= 2*periodOn)
            return (t/periodCook)*periodT + (remainderT/2.);
        else
            return (t/periodCook)*periodT + periodOn + (remainderT - periodOn*2);
    }

    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);

        k = in.nextInt();
        d = in.nextInt();
        t = in.nextInt() * 2;

        if (k <= d)
            System.out.println(calc(k, d - k));
        else
            System.out.println(calc(k, ((k + d - 1)/d)*d - k));
    }

}


import java.util.Scanner;

public class C {

    private static long k, d, t;

    private static double calc(long periodOn, long periodOff) {
        long periodT = periodOn + periodOff;
        long periodCook = periodOn*2 + periodOff;

        long remainderT = t - (t/periodCook) * periodCook;
        if (remainderT <= 2*periodOn)
            return (t/periodCook)*periodT + (remainderT/2.);
        else
            return (t/periodCook)*periodT + periodOn + (remainderT - periodOn*2);
    }

    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);

        k = in.nextLong();
        d = in.nextLong();
        t = in.nextLong() * 2;

        if (k <= d)
            System.out.println(calc(k, d - k));
        else
            System.out.println(calc(k, ((k + d - 1)/d)*d - k));
    }

}


import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.PrintWriter;
import java.io.BufferedWriter;
import java.util.InputMismatchException;
import java.io.IOException;
import java.io.Writer;
import java.io.OutputStreamWriter;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 *
 * @author cacophonix
 */
public class Main {
    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        InputReader in = new InputReader(inputStream);
        OutputWriter out = new OutputWriter(outputStream);
        TaskC solver = new TaskC();
        solver.solve(1, in, out);
        out.close();
    }

    static class TaskC {
        public void solve(int testNumber, InputReader in, OutputWriter out) {
            Thread t = new Thread(null, new Run(in, out), "name", 1 << 28);
            t.start();
            try {
                t.join();
            } catch (InterruptedException e) {

            }
        }

        public class Run extends Thread {
            InputReader in;
            OutputWriter out;

            public Run(InputReader in, OutputWriter out) {
                this.in = in;
                this.out = out;
            }

            public void run() {
                long k = in.readLong();
                long d = in.readLong();
                long t = in.readLong();

                long needed = (k + d - 1) / d;

                double single = k + (needed * d - k) / 2.0;

                double nd = t / single;

                double fl = Math.floor(nd);
                // double ce = Math.ceil(nd);

                //out.printLine(single,fl,needed,d);

                double ans = fl * needed * d;

                //out.printLine(ans);

                double com = fl * single;

                double left = t - com;

                if (left <= k) {
                    ans += left;
                } else {
                    ans += k;
                    left -= k;
                    ans += left * 2;
                }

                out.printLine(ans);

            }

        }

    }

    static class OutputWriter {
        private final PrintWriter writer;

        public OutputWriter(OutputStream outputStream) {
            writer = new PrintWriter(new BufferedWriter(new OutputStreamWriter(outputStream)));
        }

        public OutputWriter(Writer writer) {
            this.writer = new PrintWriter(writer);
        }

        public void print(Object... objects) {
            for (int i = 0; i < objects.length; i++) {
                if (i != 0) {
                    writer.print(' ');
                }
                writer.print(objects[i]);
            }
        }

        public void printLine(Object... objects) {
            print(objects);
            writer.println();
        }

        public void close() {
            writer.close();
        }

    }

    static class InputReader {
        private InputStream stream;
        private byte[] buf = new byte[1024];
        private int curChar;
        private int numChars;
        private InputReader.SpaceCharFilter filter;

        public InputReader(InputStream stream) {
            this.stream = stream;
        }

        public int read() {
            if (numChars == -1) {
                throw new InputMismatchException();
            }
            if (curChar >= numChars) {
                curChar = 0;
                try {
                    numChars = stream.read(buf);
                } catch (IOException e) {
                    throw new InputMismatchException();
                }
                if (numChars <= 0) {
                    return -1;
                }
            }
            return buf[curChar++];
        }

        public long readLong() {
            int c = read();
            while (isSpaceChar(c)) {
                c = read();
            }
            int sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = read();
            }
            long res = 0;
            do {
                if (c < '0' || c > '9') {
                    throw new InputMismatchException();
                }
                res *= 10;
                res += c - '0';
                c = read();
            } while (!isSpaceChar(c));
            return res * sgn;
        }

        public boolean isSpaceChar(int c) {
            if (filter != null) {
                return filter.isSpaceChar(c);
            }
            return isWhitespace(c);
        }

        public static boolean isWhitespace(int c) {
            return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
        }

        public interface SpaceCharFilter {
            public boolean isSpaceChar(int ch);

        }

    }
}


import java.util.Scanner;

public class Main {

	static Scanner in = new Scanner(System.in);

	public static void main(String[] args) {
		long k = in.nextLong();
		long d = in.nextLong();
		double t = in.nextDouble();

		if (k % d == 0) {
			// turn on immediately after turning off
			System.out.println(t);
			return;
		}
		double ans = 0;
		double cook = 0;

		long left = 0;

		if (k > d)
			left = d - k % d;
		else
			left = d - k;

		double oneLoop = k + left * 0.5;
		long loopTime = (long) (t / oneLoop);

		ans += loopTime * (k + left);
		cook += loopTime * oneLoop;

		if (cook + k < t) {
			cook += k;
			ans += k;
		} else {
			ans += t - cook;
			cook = t;
		}

		if (cook + left * 0.5 < t) {
			cook += left * 0.5;
			ans += left;
		} else {
			ans += (t - cook) * 2;
		}

		System.out.println(ans);
	}
}


import java.util.Scanner;

public class Main {

	static Scanner in = new Scanner(System.in);

	public static void main(String[] args) {
		long k = in.nextLong();
		long d = in.nextLong();
		double t = in.nextDouble();

		if (k % d == 0) {
			// turn on immediately after turning off
			System.out.println(t);
			return;
		}
		double ans = 0;
		double cook = 0;

		long left = 0;

		if (k > d)
			left = d - k % d;
		else
			left = d - k;

		while (cook < 1) {
			if (cook + k / t < 1) {
				cook += k / t;
				ans += k;
			} else {
				ans += (1 - cook) * t;
				cook = 1;
				break;
			}
			if (cook >= 1)
				break;

			if (cook + left * 0.5 / t < 1) {
				cook += left * 0.5 / t;
				ans += left;
			} else {
				ans += (1 - cook) * 2 * t;
				cook = 1;
				break;
			}
		}

		System.out.println(ans);
	}
}


import java.util.Scanner;

public class Main {

	static Scanner in = new Scanner(System.in);

	public static void main(String[] args) {
		long k = in.nextLong();
		long d = in.nextLong();
		double t = in.nextDouble();

		if (k % d == 0) {
			// turn on immediately after turning off
			System.out.println(t);
			return;
		}
		double ans = 0;
		double cook = 0;

		long left = 0;

		if (k > d)
			left = d - k % d;
		else
			left = d - k;

		double oneLoop = k / t + left * 0.5 / t;
		long loopTime = (long) (1.0 / oneLoop);
		ans += loopTime * (k + left);
		cook += loopTime * oneLoop;

		while (cook < 1) {
			if (cook + k / t < 1) {
				cook += k / t;
				ans += k;
			} else {
				ans += (1 - cook) * t;
				cook = 1;
				break;
			}
			if (cook >= 1)
				break;

			if (cook + left * 0.5 / t < 1) {
				cook += left * 0.5 / t;
				ans += left;
			} else {
				ans += (1 - cook) * 2 * t;
				cook = 1;
				break;
			}
		}

		System.out.println(ans);
	}
}


import java.util.Scanner;

public class Main {

	static Scanner in = new Scanner(System.in);

	public static void main(String[] args) {
		long k = in.nextLong();
		long d = in.nextLong();
		double t = in.nextDouble();

		if (k % d == 0) {
			// turn on immediately after turning off
			System.out.println(t);
			return;
		}
		double ans = 0;
		double cook = 0;

		long left = 0;

		if (k > d)
			left = d - k % d;
		else
			left = d - k;

		double oneLoop = k + left * 0.5;
		long loopTime = (long) (t / oneLoop);

		ans += loopTime * (k + left);
		cook += loopTime * oneLoop;

		while (cook < t) {
			if (cook + k  < t) {
				cook += k ;
				ans += k;
			} else {
				ans += t - cook;
				break;
			}

			if (cook + left * 0.5 < t) {
				cook += left * 0.5 ;
				ans += left;
			} else {
				ans += (t - cook) * 2;
				break;
			}
		}

		System.out.println(ans);
	}
}


import java.util.*;
import java.io.*;
import java.lang.*;

public class codes_2
	{

		public static void main(String[] args)
		{
			InputReader in = new InputReader(System.in);
			PrintWriter pw = new PrintWriter(System.out);

			//Code starts..

			double k = in.nextLong();
			double d = in.nextLong();
			double t = in.nextLong();

			double tot = Math.ceil((k/d))* d;
			double on = k/tot;
			double off = (k%d)/tot;

			//pw.println(on+" "+off);

			double one = on/t  + off/(2*t);
			double ptime = 1/one;
			ptime = Math.floor(ptime/tot)*tot;
			double ans = ptime;
			ptime = ptime * one;
			double rem = 1 - ptime;
			//pw.println(ans);
			//pw.println(rem);
			if(rem*t<=k) {
				ans += rem*t;

			}
			else
			{
				ans += k;
				rem = rem - k/t;
				ans += rem/(2*t);

			}

			pw.println(ans);
			//Code ends....

			pw.flush();
			pw.close();
		}

		static class InputReader
		{

			private final InputStream stream;
			private final byte[] buf = new byte[8192];
			private int curChar, snumChars;
			private SpaceCharFilter filter;

			public InputReader(InputStream stream)
			{
				this.stream = stream;
			}

			public int snext()
			{
				if (snumChars == -1)
					throw new InputMismatchException();
				if (curChar >= snumChars)
				{
					curChar = 0;
					try
					{
						snumChars = stream.read(buf);
					}
					catch (IOException e)
					{
						throw new InputMismatchException();
					}
					if (snumChars <= 0)
						return -1;
				}
				return buf[curChar++];
			}

			public int nextInt()
		    {
				int c = snext();
				while (isSpaceChar(c))
				{
					c = snext();
				}
				int sgn = 1;
				if (c == '-')
			    {
					sgn = -1;
					c = snext();
				}
				int res = 0;
				do
				{
					if (c < '0' || c > '9')
						throw new InputMismatchException();
					res *= 10;
					res += c - '0';
					c = snext();
				} while (!isSpaceChar(c));
				return res * sgn;
			}

			public long nextLong()
		    {
				int c = snext();
				while (isSpaceChar(c))
				{
					c = snext();
				}
				int sgn = 1;
				if (c == '-')
				{
					sgn = -1;
					c = snext();
				}
				long res = 0;
				do
				{
					if (c < '0' || c > '9')
						throw new InputMismatchException();
					res *= 10;
					res += c - '0';
					c = snext();
				} while (!isSpaceChar(c));
				return res * sgn;
			}

			public int[] nextIntArray(int n)
			{
				int a[] = new int[n];
				for (int i = 0; i < n; i++)
				{
					a[i] = nextInt();
				}
				return a;
			}

			public String readString()
		    {
				int c = snext();
				while (isSpaceChar(c))
				{
					c = snext();
				}
				StringBuilder res = new StringBuilder();
				do
				{
					res.appendCodePoint(c);
					c = snext();
				} while (!isSpaceChar(c));
				return res.toString();
			}

			public String nextLine()
			{
				int c = snext();
				while (isSpaceChar(c))
					c = snext();
				StringBuilder res = new StringBuilder();
				do
				{
					res.appendCodePoint(c);
					c = snext();
				} while (!isEndOfLine(c));
				return res.toString();
			}

			public boolean isSpaceChar(int c)
			{
				if (filter != null)
					return filter.isSpaceChar(c);
				return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
			}

			private boolean isEndOfLine(int c)
			{
				return c == '\n' || c == '\r' || c == -1;
			}

			public interface SpaceCharFilter
		    {
				public boolean isSpaceChar(int ch);
			}
		}
		public static long c = 0;

		public static long mod = 1000000007;
		public static int d;
		public static int p;
		public static int q;

		public static  int gcd(int p2, int p22)
		{
		    if (p2 == 0)
		        return (int) p22;
		    return gcd(p22%p2, p2);
		}

		public static int findGCD(int arr[], int n)
		{
		    int result = arr[0];
		    for (int i=1; i<n; i++)
		        result = gcd(arr[i], result);

		    return result;
		}

	    public static long lcm(int[] numbers) {
	        long lcm = 1;
	        int divisor = 2;
	        while (true) {
	            int cnt = 0;
	            boolean divisible = false;
	            for (int i = 0; i < numbers.length; i++) {
	                if (numbers[i] == 0) {
	                    return 0;
	                } else if (numbers[i] < 0) {
	                    numbers[i] = numbers[i] * (-1);
	                }
	                if (numbers[i] == 1) {
	                    cnt++;
	                }
	                if (numbers[i] % divisor == 0) {
	                    divisible = true;
	                    numbers[i] = numbers[i] / divisor;
	                }
	            }
	            if (divisible) {
	                lcm = lcm * divisor;
	            } else {
	                divisor++;
	            }
	            if (cnt == numbers.length) {
	                return lcm;
	            }
	        }
	    }
		public static long fact(long n) {

			long factorial = 1;
			 for(int i = 1; i <= n; i++)
		        {
		            factorial *= i;
		        }
			 return factorial;
		}

		public static int lowerLimit(int[] a, int n) {
			int ans = 0;

			int ll = 0;
			int rl = a.length-1;
		//	System.out.println(a[rl]+" "+n);
			if(a[0]>n)
				return 0;
			if(a[0]==n)
				return 1;
			else if(a[rl]<=n)
				return rl+1;

			while(ll<=rl)
			{

				int mid = (ll+rl)/2;
				if(a[mid]==n)
				{
					ans = mid + 1;
					break;
				}

				else if(a[mid]>n)
				{
					rl = mid-1;

				}
				else
				{
					ans = mid+1;
					ll = mid+1;
				}
			}

			return ans;
		}

		public static long choose(long total, long choose){
		    if(total < choose)
		        return 0;
		    if(choose == 0 || choose == total)
		        return 1;
		    return (choose(total-1,choose-1)+choose(total-1,choose))%mod;
		}

		public static int[] suffle(int[] a,Random gen)
		{
			int n = a.length;
			for(int i=0;i<n;i++)
			{
				int ind = gen.nextInt(n-i)+i;
				int temp = a[ind];
				a[ind] = a[i];
				a[i] = temp;
			}
			return a;
		}

		public static int floorSearch(int arr[], int low, int high, int x)
		{
		    if (low > high)
		        return -1;

		    if (x > arr[high])
		        return high;
		    int mid = (low+high)/2;

		    if (mid > 0 && arr[mid-1] < x && x < arr[mid])
		        return mid-1;

		    if (x < arr[mid])
		        return floorSearch(arr, low, mid-1, x);

		    return floorSearch(arr, mid+1, high, x);
		}

		public static void swap(int a, int b){
			int temp = a;
			a = b;
			b = temp;
		}
		public static ArrayList<Integer> primeFactorization(int n)
		{
			ArrayList<Integer> a =new ArrayList<Integer>();
			for(int i=2;i*i<=n;i++)
			{
				while(n%i==0)
				{
					a.add(i);
					n/=i;
				}
			}
			if(n!=1)
				a.add(n);
			return a;
		}

		public static void sieve(boolean[] isPrime,int n)
		{
			for(int i=1;i<n;i++)
				isPrime[i] = true;

			isPrime[0] = false;
			isPrime[1] = false;

			for(int i=2;i*i<n;i++)
			{
				if(isPrime[i] == true)
				{
					for(int j=(2*i);j<n;j+=i)
						isPrime[j] = false;
				}
			}
		}

		public static int GCD(int a,int b)
		{
			if(b==0)
				return a;
			else
				return GCD(b,a%b);
		}

		public static long GCD(long a,long b)
		{
			if(b==0)
				return a;
			else
				return GCD(b,a%b);
		}

		public static void extendedEuclid(int A,int B)
		{
			if(B==0)
			{
				d = A;
				p = 1 ;
				q = 0;
			}
			else
			{
				extendedEuclid(B, A%B);
				int temp = p;
				p = q;
				q = temp - (A/B)*q;
			}
		}

		public static long LCM(long a,long b)
		{
			return (a*b)/GCD(a,b);
		}

		public static int LCM(int a,int b)
		{
			return (a*b)/GCD(a,b);
		}

		public static int binaryExponentiation(int x,int n)
		{
		    int result=1;
		    while(n>0)
		    {
		        if(n % 2 ==1)
		            result=result * x;
		        x=x*x;
		        n=n/2;
		    }
		    return result;
		}

		public static int[] countDer(int n)
	    {
	        int der[] = new int[n + 1];

	        der[0] = 1;
	        der[1] = 0;
	        der[2] = 1;

	        for (int i = 3; i <= n; ++i)
	            der[i] = (i - 1) * (der[i - 1] + der[i - 2]);

	        // Return result for n
	        return der;
	    }

		 static long binomialCoeff(int n, int k)
		    {
		    long C[][] = new long[n+1][k+1];
		    int i, j;

		        // Calculate  value of Binomial Coefficient in bottom up manner
		    for (i = 0; i <= n; i++)
		    {
		        for (j = 0; j <= Math.min(i, k); j++)
		        {
		            // Base Cases
		            if (j == 0 || j == i)
		                C[i][j] = 1;

		            // Calculate value using previosly stored values
		            else
		                C[i][j] = C[i-1][j-1] + C[i-1][j];
		          }
		     }

		    return C[n][k];
		    }

		public static long binaryExponentiation(long x,long n)
		{
		    long result=1;
		    while(n>0)
		    {
		        if(n % 2 ==1)
		            result=result * x;
		        x=(x%mod * x%mod)%mod;
		        n=n/2;
		    }
		    return result;
		}

		public static int modularExponentiation(int x,int n,int M)
		{
		    int result=1;
		    while(n>0)
		    {
		        if(n % 2 ==1)
		            result=(result * x)%M;
		        x=(x*x)%M;
		        n=n/2;
		    }
		    return result;
		}

		public static long modularExponentiation(long x,long n,long M)
		{
		    long result=1;
		    while(n>0)
		    {
		        if(n % 2 ==1)
		            result=(result * x)%M;
		        x=(x*x)%M;
		        n=n/2;
		    }
		    return result;
		}

		public static int modInverse(int A,int M)
		{
		    return modularExponentiation(A,M-2,M);
		}

		public static long modInverse(long A,long M)
		{
		    return modularExponentiation(A,M-2,M);
		}

		public static boolean isPrime(int n)
		{

		    if (n <= 1)  return false;
		    if (n <= 3)  return true;

		    if (n%2 == 0 || n%3 == 0)
		    	return false;

		    for (int i=5; i*i<=n; i=i+6)
		    {
		        if (n%i == 0 || n%(i+2) == 0)
		           return false;
		    }

		    return true;
		}

		static class pair implements Comparable<pair>

		{
			Integer x, y;
			pair(int x,int y)
			{
				this.x=x;
				this.y=y;
			}

			public int compareTo(pair o) {
				int result = x.compareTo(o.x);
				if(result==0)
					result = y.compareTo(o.y);

				return result;
			}

			public String toString()
			{
				return x+" "+y;
			}

			public boolean equals(Object o)
			{
				if (o instanceof pair)
			    {
					pair p = (pair)o;
					return p.x == x && p.y == y ;
				}
				return false;
			}

			public int hashCode()
			{
				return new Long(x).hashCode()*31 + new Long(y).hashCode();
			}
		}

		static class triplet implements Comparable<triplet>
		{
			Integer x,y,z;
			triplet(int x,int y,int z)
			{
				this.x = x;
				this.y = y;
				this.z = z;
			}

			public  int compareTo(triplet o)
			{
				int result = x.compareTo(o.x);
				if(result==0)
					result =  y.compareTo(o.y);
				if(result==0)
					result = z.compareTo(o.z);

				return result;
			}

			public boolean equlas(Object o)
			{
				if(o instanceof triplet)
				{
					triplet p = (triplet)o;
					return x==p.x && y==p.y && z==p.z;
				}
				return false;
			}

			public String toString()
			{
				return x+" "+y+" "+z;
			}
			public int hashCode()
			{
				return new Long(x).hashCode()*31 + new Long(y).hashCode() + new Long(z).hashCode();
			}
		}

		static class node implements Comparable<node>

		{
			Integer x, y, z;
			node(int x,int y, int z)
			{
				this.x=x;
				this.y=y;
				this.z=z;
			}

			public int compareTo(pair o) {
				int result = x.compareTo(o.x);
				if(result==0)
					result = y.compareTo(o.y);
				if(result==0)
					result = z.compareTo(z);
				return result;
			}

			@Override
			public int compareTo(node o) {
				// TODO Auto-generated method stub
				return 0;
			}
		}
	}