
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 \( n \) be a positive integer, and let the set \( M = \{ 1, 2, \cdots, 2n \} \).

Find the smallest positive integer \( k \) such that any \( k \)-element subset of \( M \) must contain four distinct elements whose sum is \( 4n + 1 \).

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