
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
(Full score for this problem is 12 points) Given $f(x) = e^x - ax - 1$.

(1) Find the intervals where $f(x)$ is monotonically increasing.
(2) If $f(x)$ is monotonically increasing on the domain $\mathbb{R}$, find the range of possible values for $a$.
(3) Does there exist a value of $a$ such that $f(x)$ is monotonically decreasing on $(-\infty, 0]$ and monotonically increasing on $[0, +\infty)$? If so, find the value of $a$; if not, explain why.

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