
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 cheering team of Ubon Ratchathani University sits on the amphitheater that has  $441$  seats arranged into a  $21\times 21$  grid. Every seat is occupied by exactly one person, and each person has a blue sign and a yellow sign.

Count the number of ways for each person to raise one sign so that each row and column has an odd number of people raising a blue sign.

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