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.
#include <bits/stdc++.h>
using namespace std;
const int mod = 1e9 + 7;
const int M = 2e5 + 9;
int n, u, sum = 0, ans = 0;
;
int fac[M], inv[M], pw[M];
long long m, t[M];
int Calc(int y, int x) {
  if (x < y) return 0;
  return 1ll * fac[x] * inv[y] % mod * inv[x - y] % mod;
}
int main() {
  scanf("%d%lld", &n, &m);
  for (int i = 1; i <= n; ++i) scanf("%d", &t[i]);
  pw[0] = fac[1] = inv[1] = inv[0] = 1;
  pw[1] = 500000004;
  for (int i = 2; i <= n; ++i) {
    t[i] += t[i - 1];
    pw[i] = 500000004ll * pw[i - 1] % mod;
    fac[i] = 1ll * fac[i - 1] * i % mod;
    inv[i] = 1ll * (mod - mod / i) * inv[mod % i] % mod;
  }
  for (int i = 2; i <= n; ++i) inv[i] = 1ll * inv[i] * inv[i - 1] % mod;
  sum = 1;
  u = min(1ll * n, m);
  for (int i = 1; i <= n; ++i) {
    if (m < t[i]) break;
    int v = min(1ll * n, m - t[i]);
    sum = (2ll * sum % mod - Calc(u, i - 1) + mod) % mod;
    for (int j = u; j > v; --j) sum = (sum - Calc(j, i) + mod) % mod;
    u = v;
    ans = (ans + 1ll * sum * pw[i] % mod) % mod;
  }
  printf("%d", ans);
  return 0;
}
