
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 $f\left(x\right)=e^{x}-ax$, where $a \gt -1$.  
$(1)$ When $a=1$, find the equation of the tangent line to $f\left(x\right)$ at the point $\left(0,f\left(0\right)\right)$;  
$(2)$ Discuss the monotonicity of $f\left(x\right)$;  
$(3)$ Let $g\left(x\right)=f\left(x\right)-x$, find the minimum value of $g\left(x\right)$ denoted as $h\left(a\right)$, and then find the maximum value of $h\left(a\right)$.

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