
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
Given \( g_{i}(x) = a_{i} \sin \left(\omega_{i} x + \theta_{i}\right) \), where \( a_{i} \omega_{i} \neq 0 (i = 1, 2) \) and \( \left|\omega_{1}\right| \neq \left|\omega_{2}\right| \), when \( f(x) = g_{1}(x) + g_{2}(x) \) is a periodic function, it must hold that \( \frac{\omega_{1}}{\omega_{2}} \) is a rational number.

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