
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 $f(x)=\int_{-a}^{x}(12t+4a)dt$, $F(a)=\int_{0}^{1}[f(x)+3a^{2}]dx$, find the minimum value of the function $F(a)$.

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