
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 positive integer, set  $M = \{ 1, 2, \ldots, 2n \}$ . Find the minimum positive integer  $k$  such that for any subset  $A$  (with  $k$  elements) of set  $M$ , there exist four pairwise distinct elements in  $A$  whose sum is  $4n + 1$ .

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