
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  $\frac{\pi}{70}$  and  $\frac{\pi}{54}$  radians, respectively, with the positive x-axis. For any line  $l$ , the transformation  $R(l)$  produces another line as follows:  $l$  is reflected in  $l_1$ , and the resulting line is reflected in  $l_2$ . Let  $R^{(1)}(l)=R(l)$  and  $R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)$ . Given that  $l$  is the line  $y=\frac{19}{92}x$ , find the smallest positive integer  $m$  for which  $R^{(m)}(l)=l$ .

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