
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
Find the smallest positive integer \( n \) such that for any positive integer \( k \geq n \), in the set \( M = \{1, 2, \ldots, k\} \) of the first \( k \) positive integers, for any \( x \in M \), there is always another number \( y \in M \) (where \( y \neq x \)) such that \( x + y \) is a perfect square.

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