
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 $f(x)={x}^{3}+a{x}^{2}+bx+1$ with its derivative ${f}^{\mathrm{\prime }}(x)$ satisfying ${f}^{\mathrm{\prime }}(1)=2a,{f}^{\mathrm{\prime }}(2)=-b,$ where $a,b$ are constants in $\mathbb{R}$.
(1) Find the values of $a,b$ and ${f}^{\mathrm{\prime }}(x)$;
(2) Let $g(x)={f}^{\mathrm{\prime }}(x){e}^{x}$, find the extreme values of $g(x)$.

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