
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 intervals of monotonicity.

(2) Let $b\in \mathbb{R}$. If $f(x)\geqslant b$ 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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