
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
The set  $M= \{1;2;3;\ldots ; 29;30\}$  is  divided in  $k$ subsets such that if  $a+b=n^2, (a,b \in M, a\neq b, n$  is an
integer number  $)$ , then  $a$  and  $b$  belong different subsets.
Determine the minimum value of  $k$ .

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