F. Crossword Experttime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputToday 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 \frac{1}{2} he solves the crossword in exactly t_i seconds, and with probability \frac{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 = \sum \limits_{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 \frac{P}{Q} with Q > 0. To give the answer, you should print P \cdot Q^{-1} \bmod (10^9 + 7).InputThe first line contains two integers n and T (1 \le n \le 2 \cdot 10^5, 1 \le T \le 2 \cdot 10^{14}) — the number of crosswords and the time Adilbek has to spend, respectively.The second line contains n integers t_1, t_2, \dots, t_n (1 \le t_i \le 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.OutputPrint one integer — the expected value of the number of crosswords Adilbek solves in T seconds, expressed in the form of P \cdot Q^{-1} \bmod (10^9 + 7).ExamplesInput
3 5
2 2 2
Output
750000007
Input
3 5
2 1 2
Output
125000003
NoteThe answer for the first sample is equal to \frac{14}{8}.The answer for the second sample is equal to \frac{17}{8}.
3 5
2 2 2

750000007 
['combinatorics', 'dp', 'number theory', 'probabilities', 'two pointers', '*2400']