
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
Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$.

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