
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{\pi}{6}$ and $a_{n+1}=\arctan \left(\sec a_{n}\right)$ for $n \in \mathbf{N}^{*}$. Find the positive integer $m$ such that
$$
\sin a_{1} \cdot \sin a_{2} \cdots \sin a_{m}=\frac{1}{100} .
$$

Present the answer in LaTex format: \boxed{Your answer}
user