
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
If the points $A_{1}, A_{2}, A_{3}, \ldots, A_{n}$ divide the circumference of a circle into equal parts, then for any arbitrary point $P$ on the circumference:

$$
{\overline{P A_{1}}}^{2} + {\overline{P A_{2}}}^{2} + {\overline{P A_{3}}}^{2} + \cdots + {\overline{P A_{n}}}^{2} = \text{constant.}
$$

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