
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 $f(x)=e^{x}-x+ \frac {1}{2}x^{2}$ (where $e$ is the base of the natural logarithm) and the function $g(x)= \frac {1}{2}x^{2}+ax+b$ (where $a\in\mathbb{R}$ and $b\in\mathbb{R}$).

1. Find the extreme values of $f(x)$.
2. If $f(x)\geqslant g(x)$, find the maximum value of $b(a+1)$.

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