
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 a function $f(x) = \frac{1}{3}ax^3 + \frac{1}{2}bx^2 + cx$, the tangent line at the point $(-1, f(-1))$ is parallel to the x-axis, and the slope of the tangent line at the point $(1, f(1))$ is 1. Moreover, for any $x \in \mathbb{R}$, it always holds that $x \leq f'(x)$.

(Ⅰ) Find the explicit formula for $f(x)$;
(Ⅱ) Find the maximum value of $g(x) = 12f(x) - 4x^2 - 3x - 3$ on the interval $\left[\frac{1}{2}, 2\right]$;
(Ⅲ) Suppose $h(x) = \frac{m}{x} + x \cdot \ln x$, and for any $x_1, x_2 \in \left[\frac{1}{2}, 2\right]$, it holds that $h(x_1) \geq g(x_2)$. Determine the range of the real number $m$.

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