HMMT November 2020 U4

Ainsley and Buddy play a game where they repeatedly roll a standard fair six-sided die. Ainsley wins if two multiples of 3 in a row are rolled before a non-multiple of 3 followed by a multiple of 3 , and Buddy wins otherwise. If the probability that Ainsley wins is $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.