Let $ABC$ be a non-degenerate triangle with perimeter $4$ such that $a = bc \sin^2 A$. If $M$ is the maximum possible area of $\triangle ABC$ and $m$ is the minimum possible area of $\triangle ABC$, then $M^2 + m^2$ can be expressed in the form $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $a + b$.

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