
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 \(A_1B_1C_1\) with side length 10 are inscribed in the same circle such that point \(A_1\) lies on arc \(BC\) and point \(B_1\) lies on arc \(AC\). Find \(AA_1^2 + BC_1^2 + CB_1^2\).

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