Al and Bill play a game involving a fair six-sided die. The die is rolled until either a number less than $5$ is rolled on consecutive tosses, or a number greater than $4$ is rolled on consecutive tosses. Al wins if the last roll is a $5$ or $6$. Bill wins if the last roll is a $2$ or lower. Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that Bill wins. Find the value of $m+n$.

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