Today Adilbek is taking his probability theory test. Unfortunately, when Adilbek arrived at the university, there had already been a long queue of students wanting to take the same test. Adilbek has estimated that he will be able to start the test only T seconds after coming. 

Fortunately, Adilbek can spend time without revising any boring theorems or formulas. He has an app on this smartphone which contains n Japanese crosswords to solve. Adilbek has decided to solve them all one by one in the order they are listed in the app, without skipping any crossword. For each crossword, a number t_i is given that represents the time it takes an average crossword expert to solve this crossword (the time is given in seconds).

Adilbek is a true crossword expert, but, unfortunately, he is sometimes unlucky in choosing the way to solve the crossword. So, it takes him either t_i seconds or t_i + 1 seconds to solve the i-th crossword, equiprobably (with probability 1/2 he solves the crossword in exactly t_i seconds, and with probability 1/2 he has to spend an additional second to finish the crossword). All these events are independent.

After T seconds pass (or after solving the last crossword, if he manages to do it in less than T seconds), Adilbek closes the app (if he finishes some crossword at the same moment, that crossword is considered solved; otherwise Adilbek does not finish solving the current crossword at all). He thinks it would be an interesting probability theory problem to calculate E — the expected number of crosswords he will be able to solve completely. Can you calculate it? 

Recall that the expected value of a discrete random variable is the probability-weighted average of all possible values — in this problem it means that the expected value of the number of solved crosswords can be calculated as E = ∑ _{i = 0}^{n} i p_i, where p_i is the probability that Adilbek will solve exactly i crosswords. 

We can represent E as rational fraction P/Q with Q > 0. To give the answer, you should print P ⋅ Q^{-1} mod (10^9 + 7).

Input

The first line contains two integers n and T (1 ≤ n ≤ 2 ⋅ 10^5, 1 ≤ T ≤ 2 ⋅ 10^{14}) — the number of crosswords and the time Adilbek has to spend, respectively.

The second line contains n integers t_1, t_2, ..., t_n (1 ≤ t_i ≤ 10^9), where t_i is the time it takes a crossword expert to solve the i-th crossword.

Note that Adilbek solves the crosswords in the order they are given in the input without skipping any of them.

Output

Print one integer — the expected value of the number of crosswords Adilbek solves in T seconds, expressed in the form of P ⋅ Q^{-1} mod (10^9 + 7).

Examples

Input


3 5
2 2 2


Output


750000007


Input


3 5
2 1 2


Output


125000003

Note

The answer for the first sample is equal to 14/8.

The answer for the second sample is equal to 17/8.
import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.io.IOException;
import java.io.InputStreamReader;
import java.util.StringTokenizer;
import java.io.BufferedReader;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 */
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);
        FCrosswordExpert solver = new FCrosswordExpert();
        solver.solve(1, in, out);
        out.close();
    }

    static class FCrosswordExpert {
        private static final NumberTheory.Mod107 mod = new NumberTheory.Mod107();

        public void solve(int testNumber, InputReader in, PrintWriter out) {
            int n = in.nextInt();
            long T = in.nextLong();

            int[] t = in.readIntArray(n);
            long[] cumT = new long[n + 1];
            int min = -1, max = -1;
            for (int i = 0; i < n; i++) {
                cumT[i + 1] = cumT[i] + t[i];
                if (min == -1 && cumT[i + 1] + (i + 1) > T)
                    min = i;
                if (max == -1 && cumT[i + 1] > T)
                    max = i;
            }
            if (min == -1)
                min = 0;
            if (max == -1)
                max = n;

            long[] invPow2 = new long[max + 2 - min];
            long inv2 = mod.inv(2);
            invPow2[0] = mod.pow(inv2, min);
            for (int i = 0; i + 1 < invPow2.length; i++) {
                invPow2[i + 1] = mod.mult(invPow2[i], inv2);
            }

            long answer = min - 1;
            int mostBad = min;
            long pr = 0;
            for (int i = 0; i <= mostBad; i++) {
                pr = mod.add(pr, mod.ncr(min, i));
            }
            pr = mod.mult(pr, invPow2[0]);
            for (int count = min; count <= max; count++) {
                int mostBadTarget = (int) (T - cumT[count]);

                long subtract = 0;
                while (mostBad > mostBadTarget) {
                    subtract = mod.add(subtract, mod.ncr(count, mostBad));
                    mostBad--;
                }
                pr = mod.subtract(pr, mod.mult(subtract, invPow2[count - min]));

                answer = mod.add(answer, pr);

                pr = mod.subtract(pr, mod.mult(mod.ncr(count, mostBad), invPow2[count + 1 - min]));
            }

            out.println(answer);
        }

    }

    static class InputReader {
        public final BufferedReader reader;
        public StringTokenizer tokenizer;

        public InputReader(InputStream stream) {
            reader = new BufferedReader(new InputStreamReader(stream), 32768);
            tokenizer = null;
        }

        public String next() {
            while (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

        public int nextInt() {
            return Integer.parseInt(next());
        }

        public long nextLong() {
            return Long.parseLong(next());
        }

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

    }

    static class NumberTheory {
        private static void ASSERT(boolean assertion) {
            if (!assertion)
                throw new AssertionError();
        }

        public abstract static class Modulus<M extends NumberTheory.Modulus<M>> {
            final IntStack factorial = new IntStack();
            final IntStack invFactorial = new IntStack();

            public abstract long modulus();

            public Modulus() {
                super();
                factorial.push(1);
                invFactorial.push(1);
            }

            public long fact(int n) {
                while (factorial.size() <= n) {
                    factorial.push((int) mult(factorial.get(factorial.size() - 1), factorial.size()));
                }

                return factorial.get(n);
            }

            public long fInv(int n) {
                int lastKnown = invFactorial.size() - 1;

                if (lastKnown < n) {
                    long[] fInv = new long[n - lastKnown];
                    fInv[0] = inv(fact(n));
                    for (int i = 1; i < fInv.length; i++) {
                        fInv[i] = mult(fInv[i - 1], n - i + 1);
                    }
                    for (int i = fInv.length - 1; i >= 0; i--) {
                        invFactorial.push((int) fInv[i]);
                    }
                }

                return invFactorial.get(n);
            }

            public long ncr(int n, int r) {
                ASSERT(n >= 0);
                if (r < 0 || n < r)
                    return 0;
                return mult(fact(n), mult(fInv(r), fInv(n - r)));
            }

            public long normalize(long x) {
                x %= modulus();
                if (x < 0)
                    x += modulus();
                return x;
            }

            public long add(long a, long b) {
                long v = a + b;
                return v < modulus() ? v : v - modulus();
            }

            public long subtract(long a, long b) {
                long v = a - b;
                return v < 0 ? v + modulus() : v;
            }

            public long mult(long a, long b) {
                return (a * b) % modulus();
            }

            public long pow(long x, int e) {
                if (e < 0) {
                    x = inv(x);
                    e *= -1;
                }
                if (e == 0)
                    return 1;
                if ((e & 1) > 0)
                    return mult(x, pow(x, e - 1));
                return pow(mult(x, x), e / 2);
            }

            public long inv(long value) {
                long g = modulus(), x = 0, y = 1;
                for (long r = value; r != 0; ) {
                    long q = g / r;
                    g %= r;

                    long temp = g;
                    g = r;
                    r = temp;

                    x -= q * y;

                    temp = x;
                    x = y;
                    y = temp;
                }

                ASSERT(g == 1);
                ASSERT(y == modulus() || y == -modulus());

                return normalize(x);
            }

        }

        public static class Mod107 extends NumberTheory.Modulus<NumberTheory.Mod107> {
            public long modulus() {
                return 1_000_000_007L;
            }

        }

    }

    static class Util {
        public static void ASSERT(boolean assertion) {
            if (!assertion)
                throw new AssertionError();
        }

        private Util() {
        }

    }

    static class IntStack {
        int[] array;
        int size = 0;

        public IntStack() {
            this(8);
        }

        public IntStack(int capacity) {
            array = new int[capacity];
        }

        public void push(int item) {
            if (size >= array.length)
                array = resize(array);
            array[size++] = item;
        }

        public int size() {
            return size;
        }

        public int get(int i) {
            Util.ASSERT(i < size);
            return array[i];
        }

        private static int[] resize(int[] array) {
            int[] newArray = new int[array.length << 1];
            System.arraycopy(array, 0, newArray, 0, array.length);
            return newArray;
        }

        public String toString() {
            int[] trimmed = new int[size];
            System.arraycopy(array, 0, trimmed, 0, size);
            return Arrays.toString(trimmed);
        }

    }
}

