Let the domain of the function \( f(x)=\log_4\!\Big(\log_5\!\big(\log_3(18x-x^2-77)\big)\Big) \) be \( (a,b) \). Then the value of the integral \( \displaystyle \int_a^b \frac{\sin^3 x}{(\sin^3 x+\sin^3(a+b-x))}\,dx \) is equal to \( \underline{\quad} \).
