Let $ABC$ be a triangle with $BC = 49$ and circumradius $25$. Suppose that the circle centered on $BC$ that is tangent to $AB$ and $AC$ is also tangent to the circumcircle of $ABC$. Then \[\frac{AB \cdot AC}{-BC + AB + AC} = \frac{m}{n}\] where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

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