Question 24For each integer $n\gt 1,$ let $F(n)$ be the number of solutions of the equation $\sin x=\sin nx$ on the interval $[0,\pi].$ What is $\displaystyle\sum_{n=2}^{2007}F(n)?$$2{,}014{,}524$$2{,}015{,}028$$2{,}015{,}033$$2{,}016{,}532$$2{,}017{,}033$