
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
For a fixed positive integer  $n \geq 3$  we are given a  $n$   $\times$   $n$  board with all unit squares initially white. We define a *floating plus*as a  $5$ -tuple  $(M,L,R,A,B)$  of unit squares such that  $L$  is in the same row and left of  $M$ ,  $R$  is in the same row and right of  $M$ ,  $A$  is in the same column and above  $M$  and  $B$  is in the same column and below  $M$ . It is possible for  $M$  to form a floating plus with unit squares that are not next to it. Find the largest positive integer  $k$  (depending on  $n$ ) such that we can color some  $k$  squares black in such a way that there is no black colored floating plus.

*Proposed by Nikola Velov*

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