Crossword Expert

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 $$$P \cdot Q^{-1} \bmod (10^9 + 7)$$$.