Please reason step by step, and put your final answer within \boxed{}.

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 \( 4n + 1 \).
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