
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 $a\in R$, $e$ is the base of the natural logarithm.
(1) When $a=-1$, find the equation of the tangent line to the function $f(x)$ at the point $(1,f(1))$;
(2) Discuss the monotonicity of the function $f(x)$, and write out the corresponding monotonic intervals;
(3) Given $b\in R$, if the function $f(x)\geqslant b$ holds for any $x\in R$, find the maximum value of $ab$.

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