
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 \( k \) be a positive integer. Each cell in a \( 3k \times 3k \) grid is colored either red or blue. A triplet of cells \( (C_1, C_2, C_3) \) is called a "good cell group" if \( C_1 \) and \( C_2 \) are in the same row, \( C_2 \) and \( C_3 \) are in the same column, and \( C_1 \) and \( C_3 \) are colored blue while \( C_2 \) is colored red. Find the maximum possible number of good cell groups.

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