
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 $f\left(x\right)=(e^{x}-1)\sin x, x\in \left(0,2\pi \right)$.  
$(1)$ Find the equation of the tangent line to $f\left(x\right)$ at the point $P\left(\pi ,f\left(\pi \right)\right)$;  
$(2)$ Let $g\left(x\right)=f\left(x\right)-x^{2}$, $x\in \left(0,2\pi \right)$. Determine the number of zeros of $g\left(x\right)$ and explain the reason.

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