
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 $f(x) = 2x^3 - ax^2 + 8$.

(1) If $f(x) < 0$ holds for all $x \in [1, 2]$, find the range of the real number $a$.

(2) Determine whether there exists an integer $a$ such that the function $g(x) = f(x) + 4ax^2 - 12a^2x + 3a^3 - 8$ has a local minimum in the interval $(0, 1)$. If it exists, find all such integer values of $a$; if not, explain why.

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