
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 $a > 0$, the functions $f(x) = a^2 x^3 - 3ax^2 + 2$ and $g(x) = -3ax + 3$.
(1) If $a = 1$, find the equation of the tangent to the graph of $f(x)$ at the point where $x = 1$;
(2) Find the extremum of the function $f(x)$ in the interval $[-1, 1]$;
(3) If there exists an $x_0 \in (0, \frac{1}{2}]$ such that the inequality $f(x_0) > g(x_0)$ holds true, find the range of values for $a$.

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