
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 $f(x) = e^x - ax + a$, where $a \in \mathbb{R}$ and $e$ is the base of the natural logarithm:
1. Discuss the monotonicity of the function $f(x)$ and write down the corresponding monotonic interval(s).
2. Let $b \in \mathbb{R}$. If the function $f(x) \geq b$ holds for all $x \in \mathbb{R}$, find the maximum value of $ab$.

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