
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

Let \( n = 34000 \). Among the vertices of a regular \( n \)-gon \( A_{1} A_{2} \ldots A_{n} \), the vertices \( A_{i} \) are painted red if the index \( i \) is a power of 2, i.e., \( i = 1, 2, 4, 8, 16, \ldots \). In how many ways can you select 400 vertices of this \( n \)-gon such that they form a regular 400-gon and none of them are red?

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