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

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

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