
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

On some planet, there are \(2^{N}\) countries \((N \geq 4)\). Each country has a flag \(N\) units wide and one unit high composed of \(N\) fields of size \(1 \times 1\), each field being either yellow or blue. No two countries have the same flag.

We say that a set of \(N\) flags is diverse if these flags can be arranged into an \(N \times N\) square so that all \(N\) fields on its main diagonal will have the same color. Determine the smallest positive integer \(M\) such that among any \(M\) distinct flags, there exist \(N\) flags forming a diverse set.

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