
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 functions $f(x)=\ln(x+1)$ and $g(x)=e^{x}-1$.

$(1)$ Determine the number of zeros of the function $h(x)=f(x)-g(x)$.

$(2)$ Compare $\ln(e^{2}-\ln 2)$, $2-f(\ln 2)$, and $g(e^{2}-\ln 2-1)$, and explain the reasoning.

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