
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 $f(x)=e^{x}-ax-2$, where $a$ is a constant.

$(1)$ Find the intervals of monotonicity for the function $f(x)$;

$(2)$ If $y=e^x-2$ is a tangent line of $f(x)=e^{x}-ax-2$, find the value of $a$;

$(3)$ Given $a=1$ and $k$ is an integer, if for any $x\in(0,+\infty)$, $(x-k)f'(x)+x+1 > 0$ always holds, find the maximum value of $k$.

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