2025 AIME II Problem 9

There are n values of x in the interval 0<x<2\pi where f(x)=\sin(7\pi\cdot\sin(5x))=0. For t of these n values of x, the graph of y=f(x) is tangent to the x-axis. Find n+t.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.