
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
Let $f(x)=e^{x}-ax-2$.  
$(1)$ If the slope of the tangent line to the curve $y=f(x)$ at the point $(0,f(0))$ is $1$, find the value of the real number $a$.  
$(2)$ Find the monotonic interval of $f(x)$.  
$(3)$ If $a=1$, $k$ is an integer, and for $x>0$, $(x-k)f'(x)+x+1>0$ always holds, find the maximum value of $k$.

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