A wall made of mirrors has the shape of $\triangle ABC$, where $AB = 13$, $BC = 16$, and $CA = 9$. A laser positioned at point $A$ is fired at the midpoint $M$ of $BC$. The shot reflects about $BC$ and then strikes point $P$ on $AB$. If $\frac{AM}{MP} = \frac{m}{n}$ for relatively prime positive integers $m, n$, compute $100m+n$.

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