
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 $f(x)=f'(1)e^{x-1}-f(0)x+\frac{1}{2}x^{2}(f′(x) \text{ is } f(x))$'s derivative, where $e$ is the base of the natural logarithm, and $g(x)=\frac{1}{2}x^{2}+ax+b(a\in\mathbb{R}, b\in\mathbb{R})$
(I) Find the analytical expression and extreme values of $f(x)$;
(II) If $f(x)\geqslant g(x)$, find the maximum value of $\frac{b(a+1)}{2}$.

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