Crossword Expert

题意翻译

你有 $n$ 个题目 $T$ 秒时间，每个题目需要消耗时间 $t_i$ 秒。 但每题有 $50\%$ 的概率消耗多 $1$ 秒，问从 $1$ 开始一道一道向后做，$T$ 秒时做出题目数目的期望值。

题目描述

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 $ \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) $ .

输入输出格式

输入格式

The 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.

输出格式

Print 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) $ .

输入输出样例

输入样例 #1

3 5
2 2 2

输出样例 #1

750000007

输入样例 #2

3 5
2 1 2

输出样例 #2

125000003

说明

The answer for the first sample is equal to $ \frac{14}{8} $ . The answer for the second sample is equal to $ \frac{17}{8} $ .

题意翻译

你有 $n$ 个题目 $T$ 秒时间，每个题目需要消耗时间 $t_i$ 秒。 但每题有 $50\%$ 的概率消耗多 $1$ 秒，问从 $1$ 开始一道一道向后做，$T$ 秒时做出题目数目的期望值。