March Break 2024 - Problem 12

A 7-meter long graham cracker is being eaten by Aqua and Blue. Aqua and Blue take turns, starting with Aqua, eating the graham cracker in the following way:

• On each of Aqua’s turns, there is a \tfrac12 chance that they eat one meter of the graham cracker, and a \tfrac12 chance they do not take a bite.
• On each of Blue’s turns, there is a \tfrac12 chance, independent of Aqua, that Blue eats two meters or the entire cracker, whichever is smaller, and a \tfrac12 chance they do not take a bite.

If the probability that Aqua finishes eating the cracker can be expressed as \frac{a}{b}, where a and b are relatively prime, positive integers, find the remainder when a+b is divided by 1000.