
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
Each of the  $n^2$  cells of an  $n \times n$  grid is colored either black or white. Let  $a_i$  denote the number of white cells in the  $i$ -th row, and let  $b_i$  denote the number of black cells in the  $i$ -th column. Determine the maximum value of  $\sum_{i=1}^n a_ib_i$  over all coloring schemes of the grid.

*Proposed by Alex Zhai*

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