
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
Lines $l_1$ and $l_2$ pass through the origin with first-quadrant angles of $\frac{\pi}{100}$ and $\frac{\pi}{60}$, respectively, with the positive $x$-axis. For any line $l$, the transformation $R(l)$ produces another line by first reflecting $l$ in $l_1$ and then reflecting the result in $l_2$. Let $R^{(1)}(l) = R(l)$ and $R^{(n)}(l) = R(R^{(n-1)}(l))$. Given that $l$ is the line $y=\frac{15}{101}x$, find the smallest positive integer $m$ for which $R^{(m)}(l)=l$.

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