
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 cell of an $n \times n$ square board is painted either blue or red. A row or column is called "bluish" if it contains more blue cells than red ones, and it is called "reddish" if it contains more red cells than blue ones. What is the maximum value that the sum of the number of reddish rows and the number of bluish columns can take for some coloring of the board depending on $n$?

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