
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
nic $\kappa$ y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers  $d>0$  such that for every positive integer  $n,$  nic $\kappa$ y can label at least  $dn^2$  cells of an  $n\times n$  square.
 
*Proposed by Mihir Singhal and Michael Kural*


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