
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 $f(x) = f'(1)e^{x-1} - f(0)x + \frac{1}{2}x^2$, where $f'(x)$ is the derivative of $f(x)$, and $e$ is the base of the natural logarithm. Also given is the function $g(x) = \frac{1}{2}x^2 + ax + b$, where $a, b \in \mathbb{R}$.

1. Find the analytical expression of $f(x)$ and its extreme values.
2. If $f(x) \geq g(x)$, find the maximum value of $\frac{b(a+1)}{2}$.

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