Let sequence $\{a_n\}$ satisfy the condition: $a_1=\frac{\pi}{6}$ and $a_{n+1}=\arctan(\sec a_n)$, where $n\in Z^+$. There exists a positive integer $m$ such that $\sin{a_1}\cdot\sin{a_2}\cdots\sin{a_m}=\frac{1}{100}$. Find $m$.