AIME 2026b Problem 7

A standard fair six-sided die is rolled repeatedly. Each time the die reads 1 or 2, Alice gets a coin; each time it reads 3 or 4, Bob gets a coin; and each time it reads 5 or 6, Carol gets a coin. The probability that Alice and Bob each receive at least two coins before Carol receives any coins can be written as $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $100m+n.$