
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
Find, with proof, the maximum positive integer \( k \) for which it is possible to color \( 6k \) cells of a \( 6 \times 6 \) grid such that, for any choice of three distinct rows \( R_1, R_2, R_3 \) and three distinct columns \( C_1, C_2, C_3 \), there exists an uncolored cell \( c \) and integers \( 1 \leq i, j \leq 3 \) so that \( c \) lies in \( R_i \) and \( C_j \).

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