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$\large\displaystyle\int_{0}^{\frac{\pi}{2}}\ln^{2}(\sin(x))\mathrm dx=\frac{{\pi}^{3}}{a}+\frac{\pi}{b}\ln^{2}{(c)}$ If the equation above is true for constants $a,b,c$, find $a+b+c$.

This is part of Ordered Disorder.

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