
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
Given an odd number \( n \geq 3 \), color each cell of an \( n \times n \) grid with two colors, black and white. Define two cells that share a common vertex and have the same color as "adjacent." For any two cells \( a \) and \( b \), if there exists a sequence of cells \( c_1, c_2, \ldots, c_k \) such that \( c_1 = a \), \( c_k = b \), and \( c_i (i = 1, 2, \ldots, k-1) \) is adjacent to \( c_{i+1} \), then \( a \) and \( b \) are called "connected." Find the largest positive integer \( M \) such that there exists a coloring scheme where there are \( M \) cells that are pairwise not connected.

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