Suppose a biased coin gives head with probability $\frac{2}{3}$. 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 gets a $6$, and that player is declared the winner.\n\nIf the probability that $A$ will win is given by $\frac{m}{n}$, where $m$ and $n$ are coprime, then what is the value of $m^2n$?

Let's think step by step and output the final answer within \boxed{}.
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