
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 $l_2^{}$ each pass through the origin with first-quadrant angles of $\frac{\pi}{50}$ and $\frac{\pi}{45}$ with the positive $x$-axis, respectively. For any line $l$, the transformation $R(l)$ is defined in the following way: $l$ is reflected in $l_1$, and then the resulting line is reflected in $l_2$. Define $R^{(1)}(l)=R(l)$ and $R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)$. Given that $l$ is the line $y=\frac{21}{88}x$, find the smallest positive integer $n$ such that $R^{(n)}(l) = R(l)$.

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