
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
\( f(x) \) is an odd function, \( g(x) \) is an even function, and \( f(x) + g(x) + f(x) g(x) = \left(e^x + \sin x\right) \cdot \cos x \). Determine the number of zeros of \( f(x) \) in the interval \([0, 4\pi)\).

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