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Found problems: 1

STEMS 2023 Math Cat A, 7
Suppose a biased coin gives head with probability $\dfrac{2}{3}$. The coin is tossed repeatedly, if it shows heads then player $A$ rolls a fair die, otherwise player $B$ rolls the same die. The process ends when one of the players get a $6$, and that player is declared the winner. If the probability that $A$ will win is given by $\dfrac{m}{n}$ where $m,n$ are coprime, then what is the value of $m^2n$?
computational coin toss probability algebra number theory