
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.

*Proposed by Tonći Kokan, Croatia*

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