You are developing frog-shaped robots, and decided to race them against each other.

First, you placed N robots onto a number line. These robots are numbered 1 through N. The current coordinate of robot i is x_i. Here, all x_i are integers, and 0 < x_1 < x_2 < ... < x_N.

You will repeatedly perform the following operation:

* Select a robot on the number line. Let the coordinate of the robot be x. Select the destination coordinate, either x-1 or x-2, that is not occupied by another robot. The robot now jumps to the selected coordinate.



When the coordinate of a robot becomes 0 or less, the robot is considered finished and will be removed from the number line immediately. You will repeat the operation until all the robots finish the race.

Depending on your choice in the operation, the N robots can finish the race in different orders. In how many different orders can the N robots finish the race? Find the answer modulo 10^9+7.

Constraints

* 2 ≤ N ≤ 10^5
* x_i is an integer.
* 0 < x_1 < x_2 < ... < x_N ≤ 10^9

Input

The input is given from Standard Input in the following format:


N
x_1 x_2 ... x_N


Output

Print the number of the different orders in which the N robots can finish the race, modulo 10^9+7.

Examples

Input

3
1 2 3


Output

4


Input

3
2 3 4


Output

6


Input

8
1 2 3 5 7 11 13 17


Output

10080


Input

13
4 6 8 9 10 12 14 15 16 18 20 21 22


Output

311014372
import java.io.*;
import java.util.Arrays;
import java.util.StringJoiner;
import java.util.StringTokenizer;

public class Main {

    static int N;
    static int[] X;

    public static void main(String[] args) {
        FastScanner sc = new FastScanner(System.in);
        N = sc.nextInt();
        X = sc.nextIntArray(N);

        System.out.println(solve());
    }

    static int solve() {
        int ans = 1;
        int prev = 0; // 前にいる駒の数
        int order = 1;
        for (int i = 0; i < N; i++) {
            int x = X[i];
            int space = x - 1;
            boolean pittari;
            if( space % 2 == 0 ) {
                // 奇数でpittariが発生するのでここでは起きない
                pittari = false;

            } else {
                // |o-o-ox pittari
                pittari = (space - prev) == prev-1;
            }

            if( pittari ) {
                // debug(order, prev+1);
                ans = mul(ans, prev+1);
                // 減って増える
                // prev-- prev++
                order++;

            } else {
                prev++;
            }
        }

        // 残りは全部できる感じなのでそのようにする
        for (int i = order; i <= N; i++) {
            // debug(i, N-i+1);
            ans = mul(ans, N-i+1);
        }
        return ans;
    }

    static int MOD = 1_000_000_007;

    static int pow(int base, long exp) {
        if( exp == 0 ) return 1;

        int ans = 1;
        base %= MOD;
        while( exp > 0 ) {
            if( (exp & 1) == 1 ) {
                ans = mul(ans, base);
            }

            base = mul(base, base);
            exp = exp >> 1;
        }
        return ans;
    }

    static int sub(int a, int b) {
        int c = a - b;
        if( c < 0 ) c += MOD;
        return c;
    }

    static int div(int a, int b) {
        return mul(a, pow(b, MOD-2));
    }

    static int add(int a, int b) {
        int c = a + b;
        if( c >= MOD ) c %= MOD;
        return c;
    }

    static int mul(int a, int b) {
        long c = (long)a * b;
        if( c >= MOD ) c %= MOD;
        return (int)c;
    }

    @SuppressWarnings("unused")
    static class FastScanner {
        private BufferedReader reader;
        private StringTokenizer tokenizer;

        FastScanner(InputStream in) {
            reader = new BufferedReader(new InputStreamReader(in));
            tokenizer = null;
        }

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

        String nextLine() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    return reader.readLine();
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken("\n");
        }

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

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

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

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

        long[] nextLongArray(int n) {
            long[] a = new long[n];
            for (int i = 0; i < n; i++) a[i] = nextLong();
            return a;
        }
    }

    static void writeLines(int[] as) {
        PrintWriter pw = new PrintWriter(System.out);
        for (int a : as) pw.println(a);
        pw.flush();
    }

    static void writeLines(long[] as) {
        PrintWriter pw = new PrintWriter(System.out);
        for (long a : as) pw.println(a);
        pw.flush();
    }

    static void writeSingleLine(int[] as) {
        PrintWriter pw = new PrintWriter(System.out);
        for (int i = 0; i < as.length; i++) {
            if (i != 0) pw.print(" ");
            pw.print(as[i]);
        }
        pw.println();
        pw.flush();
    }

    static int max(int... as) {
        int max = Integer.MIN_VALUE;
        for (int a : as) max = Math.max(a, max);
        return max;
    }

    static int min(int... as) {
        int min = Integer.MAX_VALUE;
        for (int a : as) min = Math.min(a, min);
        return min;
    }

    static void debug(Object... args) {
        StringJoiner j = new StringJoiner(" ");
        for (Object arg : args) {
            if (arg == null) j.add("null");
            else if (arg instanceof int[]) j.add(Arrays.toString((int[]) arg));
            else if (arg instanceof long[]) j.add(Arrays.toString((long[]) arg));
            else if (arg instanceof double[]) j.add(Arrays.toString((double[]) arg));
            else if (arg instanceof Object[]) j.add(Arrays.toString((Object[]) arg));
            else j.add(arg.toString());
        }
        System.err.println(j.toString());
    }

    static void printSingleLine(int[] array) {
        PrintWriter pw = new PrintWriter(System.out);
        for (int i = 0; i < array.length; i++) {
            if (i != 0) pw.print(" ");
            pw.print(array[i]);
        }
        pw.println();
        pw.flush();
    }

    static int lowerBound(int[] array, int value) {
        int lo = 0, hi = array.length, mid;
        while (lo < hi) {
            mid = (hi + lo) / 2;
            if (array[mid] < value) lo = mid + 1;
            else hi = mid;
        }
        return lo;
    }

    static int upperBound(int[] array, int value) {
        int lo = 0, hi = array.length, mid;
        while (lo < hi) {
            mid = (hi + lo) / 2;
            if (array[mid] <= value) lo = mid + 1;
            else hi = mid;
        }
        return lo;
    }
}
