
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 quadratic function $f(x)=ax^{2}+bx$ ($a\neq 0, a, b$ are constants) that satisfies $f(1-x)=f(1+x)$, and the equation $f(x)=2x$ has two equal real roots. Let $g(x)= \frac{1}{3}x^{3}-x-f(x)$.

(I) Find the analytical expression of $f(x)$;
(II) Find the maximum value of $g(x)$ on $[0,3]$.

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