Let $a$, $b$, $c$, and $d$ be positive real numbers such that
\[ a^2 + b^2 - c^2 - d^2 = 0 \] and \[ a^2 - b^2 - c^2 + d^2 = \frac{56}{53}(bc + ad). \]
Let $M$ be the maximum possible value of $\frac{ab+cd}{bc+ad}$. If $M$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $100m + n$.

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