GAIC Math 2024 Problem 57

Let $a_1=\frac{\pi}{6}, a_n \in (0,\frac{\pi}{2})$, and $\tan a_{n+1}\cdot \cos a_n=1 (n\geq 1)$. If $\prod\limits_{k=1}^m \sin a_k=\frac{1}{100}$, then $m=$ _____.