
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 with first-quadrant angles $\frac{\pi}{80}$ and $\frac{\pi}{60}$ radians relative to the positive $x$-axis, respectively. When any line $l$ undergoes the transformation $R(l)$, it follows this procedure: $l$ is first reflected across $l_1$ and then the resulting line is reflected across $l_2$. Define $R^{(1)}(l) = R(l)$ and $R^{(n)}(l)=R(R^{(n-1)}(l))$ for any positive integer $n$. If $l$ is the line defined by the equation $y=\frac{17}{88}x$, determine the smallest positive integer $m$ such that $R^{(m)}(l) = l$.

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