Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$ respectively. Compute $MN^2 - PQ^2$.

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