
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
Partition the set \( S = \{1, 2, \ldots, 36\} \) into \( k \) pairwise disjoint non-empty subsets \( A_1, A_2, \ldots, A_k \) such that for each \( A_i \) (where \( 1 \leq i \leq k \)), the sum of any two distinct elements is not a perfect square. Find the minimum possible value of \( k \).

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