
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 the function be $f(x) = e^x - ax - 2$.  
(1) Find the intervals of monotonicity for $f(x)$.  
(2) If $a = 1$ and $k$ is an integer, and when $x > 0$, $(x - k)f'(x) + x + 1 > 0$, find the maximum value of $k$.

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