
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 the function $f(x) = e^{x} - \lambda \sin x$.  
$(1)$ Find the largest integer $\lambda$ such that for any $x \in (0, \pi)$, $f(x) > 0$;  
$(2)$ If the function $g(x) = e^{x} + 3\sin x - 1 - ax$, when $1 \leq a < 4$, discuss the number of zeros of the function $g(x)$.  
Reference data: $e^{\frac{π}{4}} \approx 2.193$

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