
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
The cells of a $40 \times 40$ table are colored in $n$ colors such that for any cell, all $n$ colors can be found within the union of its row and column. Find the maximum possible number of blue cells if:

(a) $n=2$;

(b) $n=20$.

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