
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
For a polygon \(A_{1} A_{2} \ldots A_{n}\), the area of the polygon formed by the points dividing the sides \(A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{n} A_{1}\) in the same ratio \(\lambda\) is minimized at what value of \(\lambda\)?

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