
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 $f(x) = ax^3 + bx$ ($x \in \mathbb{R}$), and $g(x) = f(x) + 3x - x^2 - 3$, and $t(x) = \frac{c}{x^2} + \ln x$:
  
  (I) If the tangent line of the graph of $f(x)$ at the point $x = 3$ is parallel to the line $24x - y + 1 = 0$, and $f(x)$ has an extreme value at $x = 1$, find the analytic expression for $f(x)$ and determine the interval where $f(x)$ is monotonically decreasing.
  
  (II) Under the conditions of (I), if for any $x_1, x_2 \in \left[\frac{1}{3}, 2\right]$, the inequality $x_1 \cdot t(x_1) \geq g(x_2)$ holds, find the range of values for the real number $c$.

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