
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 $l_2^{}$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{30}$ and $\frac{\pi}{20}$ radians, respectively, with the positive $x$-axis. A new line $l$ is given by $y=\frac{1}{11}x$. Determine the smallest positive integer $m$ for which $R^{(m)}(l)=l$, where $R(l)$ is defined as before: $l$ is reflected across $l_1$, then this result is reflected across $l_2$.

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