
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 $f\left(x\right)=\lambda \ln \left(x+1\right)-\sin x$.  
$(1)$ If $f\left(x\right)$ has a period of $2\pi$ on $\left(0,+\infty \right)$, find the value of $\lambda$.  
$(2)$ When $\lambda =1$, determine the number of zeros of the function $f\left(x\right)$ on $[\frac{π}{2},+∞)$.  
$(3)$ Given $f\left(x\right)\geqslant 2(1-e^{x})$ holds for $x\in \left[0,\pi \right]$, find the range of real numbers for $\lambda$.

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