
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 how many ways every unit square of a  $2018$  x  $2018$  board can be colored in red or white such that number of red unit squares in any two rows are distinct and number of red squares in any two columns are distinct.

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