
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 $f\left(x\right)=e^{x}-1$.<br/>$(1)$ If $g\left(x\right)=f\left(x\right)-ax\left(a\in R\right)$, discuss the monotonicity of $g\left(x\right)$.<br/>$(2)$ When $x \gt 0$, the inequality $\left(x-k-1\right)f\left(x\right)+x+1 \gt 0$ holds, find the maximum value of the integer $k$.

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