In $\triangle ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. $M$ is the midpoint of side $AB$, and $H$ is the foot of the altitude from $A$ to $BC$. If the length of $HM$ is $x$, find the value of $10x$.

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