
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 $f(x)=\frac{1}{3}x^{3}+\frac{a-3}{2}x^{2}+(a^{2}-3a)x-2a$,

(1) If for any $x \in [1,2]$, $f'(x) > a^{2}$ always holds, find the range of values for $a$;

(2) Let the two extreme points of the function $f(x)$ be $x_{1}$ and $x_{2}$, find the minimum value of $g(a)=x_{1}^{3}+x_{2}^{3}+a^{3}$.

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