Let the domain of the function $f(x)=\log _4\left(\log _5\left(\log _3\left(18 x-x^2-77\right)\right)\right)$ be $(a, b)$. Then the value of the integral $\int_a^b \frac{\sin ^3 x}{\left(\sin ^3 x+\sin ^3(a+b-x)\right)} d x$ is equal to $\_\_\_\_$ .
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