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

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 *
 * @author Rustam Musin (t.me/musin_acm)
 */
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);
        FEkspertPoKrossvordam solver = new FEkspertPoKrossvordam();
        solver.solve(1, in, out);
        out.close();
    }

    static class FEkspertPoKrossvordam {
        int mod = (int) 1e9 + 7;
        int[] fact;
        int[] inv;
        int[] factInv;
        int[] pow2inv;

        public void solve(int testNumber, InputReader in, OutputWriter out) {
            int n = in.readInt();
            long t = in.readLong();
            init(Math.max(n, 2));
            int curSum = 1;
            int curK = 0;
            int res = 0;
            long prevt = 0;
            for (int i = 0; i < n; i++) {
                curSum = add(mul(curSum, 2), c(i, ++curK));
                prevt += in.readInt();
                if (prevt > t) {
                    break;
                }
                int possibleFails = (int) Math.min(i + 1, t - prevt);
                while (curK < possibleFails) {
                    curSum = add(curSum, c(i + 1, ++curK));
                }
                while (curK > possibleFails) {
                    curSum = sub(curSum, c(i + 1, curK--));
                }
                res = add(res, mul(curSum, pow2inv[i + 1])); //sumc (1, ...)
            }
            out.print(res);
        }

        int add(int a, int b) {
            a += b;
            if (a >= mod) {
                a -= mod;
            }
            return a;
        }

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

        int mul(int a, int b) {
            long c = a * (long) b;
            return (int) (c % mod);
        }

        int norm(int x) {
            x %= mod;
            x += mod;
            x %= mod;
            return x;
        }

        void init(int n) {
            fact = new int[n + 1];
            inv = new int[n + 1];
            factInv = new int[n + 1];
            pow2inv = new int[n + 1];
            fact[0] = inv[0] = factInv[0] = pow2inv[0] = 1;
            fact[1] = inv[1] = 1;
            for (int i = 2; i <= n; i++) {
                fact[i] = mul(fact[i - 1], i);
                inv[i] = norm(mul(-mod / i, inv[mod % i]));
            }
            for (int i = 1; i <= n; i++) {
                factInv[i] = mul(factInv[i - 1], inv[i]);
                pow2inv[i] = mul(pow2inv[i - 1], inv[2]);
            }
        }

        int c(int n, int k) {
            if (k > n) {
                return 0;
            }
            return mul(fact[n], mul(factInv[k], factInv[n - k]));
        }

    }

    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 int readInt() {
            int c = read();
            while (isSpaceChar(c)) {
                c = read();
            }
            int sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = read();
            }
            int 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 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);

        }

    }

    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 close() {
            writer.close();
        }

        public void print(int i) {
            writer.print(i);
        }

    }
}

