Pinball chute bounces

Find the expected number of ticks a pinball takes to fall out of a chute, as a value modulo 1,000,000,007.

A chute has n slots in a vertical line, numbered 1 (top) to n (bottom). A pinball starts in slot start. Every slot has a peg, and pegs[i - 1] = [a, b] describes the peg in slot i. Each tick, the ball in slot i is knocked by that slot's peg:

• with probability a / b it drops one slot down; from slot n that means it leaves the chute and the game is over;
• otherwise (probability 1 - a / b) it is knocked one slot up; from slot 1 there is no slot above, so it bounces off the ceiling and stays in slot 1.

The tick on which the ball drops out of slot n counts. Every peg has 1 <= a <= b, so each slot drops the ball with positive probability and the ball always gets out eventually. The expected number of ticks is a fraction P / Q; it can be astronomically large when pegs push mostly upward, so report it as P * Q^(-1) modulo 1,000,000,007, where Q^(-1) is the modular inverse. It is guaranteed that Q is not divisible by 1,000,000,007.

Input:

• pegs: a non-empty list of [a, b] pairs, one per slot from top to bottom.
• start: the starting slot, 1 <= start <= pegs.length.

Output: an integer from 0 to 1,000,000,006, the expected number of ticks modulo 1,000,000,007.

Setting up one equation per slot and solving the whole system the long way is too slow for a long chute; look for how the equations chain together.