
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 $f(x)=e^{x}-a(x-1)$, where $a > 0$, and $e$ is the base of the natural logarithm.
$(1)$ Find the intervals of monotonicity for the function $f(x)$;
$(2)$ Given $b\in\mathbb{R}$, if the function $f(x)\geqslant b$ holds for any $x\in\mathbb{R}$, find the maximum value of $ab$.

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