
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
In a  $999 \times 999$  square table some cells are white and the remaining ones are red. Let  $T$  be the number of triples  $(C_1,C_2,C_3)$  of cells, the first two in the same row and the last two in the same column, with  $C_1,C_3$  white and  $C_2$  red. Find the maximum value  $T$  can attain.

*Proposed by Merlijn Staps, The Netherlands*

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