
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 $f(x)=e^{x}-ax-2$.  
$(1)$ Find the monotonic interval of $f(x)$;  
$(2)$ If $a=1$ and when $x>0$, $(x-k)f'(x)+x+1>0$, find the maximum value of the integer $k$.

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