
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 $f(x) = 2x^3 - 3(a+1)x^2 + 6ax$, where $a \in \mathbb{R}$.

(1) Find the value of $a$ such that the slope of the tangent line to the curve $y = f(x)$ at $x = 0$ is 3.

(2) Find the range of $a$ such that $f(x) + f(-x) \geq 12\ln x$ holds for any $x \in (0, +\infty)$.

(3) If $a > 1$, let $M(a)$ and $m(a)$ be the maximum and minimum values of the function $f(x)$ on the interval $[1, 2]$, respectively. Define $h(a) = M(a) - m(a)$. Find the minimum value of $h(a)$.

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