
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 angles of \( \frac{\pi}{60} \) and \( \frac{\pi}{48} \) radians, respectively, with the positive \( x \)-axis. For any line \( l \), the transformation \( R(l) \) is defined as follows: \( l \) is reflected in \( l_1 \), and the resulting line is reflected in \( l_2 \). Let \( R^{(1)}(l) = R(l) \) and \( R^{(n)}(l) = R(R^{(n-1)}(l)) \). Given that \( l \) is the line \( y = \frac{17}{89}x \), find the smallest positive integer \( m \) for which \( R^{(m)}(l) = l \).

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