
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
Let  $n \geq 3$  be an odd number and suppose that each square in a  $n \times n$  chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares  $a,b$  are considered connected if there exists a sequence of squares  $c_1,\ldots,c_k$  with  $c_1 = a, c_k = b$  such that  $c_i, c_{i+1}$  are adjacent for  $i=1,2,\ldots,k-1$ . 


Find the maximal number  $M$  such that there exists a coloring admitting  $M$  pairwise disconnected squares. 

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