
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 $f(x) = \frac{1}{3}x^3 - \frac{a+1}{2}x^2 + ax - 1$ and $g(x) = \frac{1}{2}(a-4)x^2$, where $a \geq 1$.
(I) The range of $f(x)$ on the interval $(0,2)$ is $(s,t)$. Find the range of values for $a$.
(II) If $a \geq 3$, then for any two distinct real numbers $x_1$, $x_2$ in the interval $[2, 3]$, the inequality $|f(x_1) - f(x_2)| > |g(x_1) - g(x_2)|$ holds. Determine the range of values for the real number $a$.

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