AIME II 2025 (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