
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 a convex polygon \(A_{1} A_{2} \ldots A_{n}\) with area \(S\), and a point \(M\) in the same plane, determine the area of polygon \(M_{1} M_{2} \ldots M_{n}\), where \(M_{i}\) is the image of \(M\) under rotation \(\mathcal{R}_{A}^{\alpha}\) around \(A_{i}\) by \(\alpha\), \(i = 1, 2, \ldots, n\).

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