
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]


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Let \(P\) be any point inside a circle \(O\) with radius \(R\). From point \(P\), draw \(n\) lines \(l_{1}, l_{2}, \cdots, l_{n}\) (\(n \geqslant 2\)). If the angle between each pair of adjacent lines is \(\frac{\pi}{n}\), and the \(i\)-th line \(l_{i}\) intersects the circle at points \(M_{i}\) and \(M_{i}'\) (\(i=1,2, \cdots, n\)), then \(\sum_{i=1}^{n}\left(\left|P M_{i}\right|^{2}+\left|P M_{i}'\right|^{2}\right) = 2 n R^{2}\), which is a constant independent of the position of point \(P\).

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