
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 $f(x)=e^{x}(ax+b)+x^{2}+2x$, the curve $y=f(x)$ passes through point $P(0,1)$, and the tangent line at point $P$ is $l$: $y=4x+1$.  
$(I)$ Find the values of $a$ and $b$;  
$(II)$ If there exists a real number $k$, such that for $x\in[-2,-1]$, $f(x)\geqslant x^{2}+2(k+1)x+k$ always holds, find the range of values for $k$.

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