
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
There are 10 boys and 10 girls forming two columns for a group dance. 

(1) If boys and girls are each in separate columns, how many different arrangements are there?

(2) If boys and girls can stand in any column, each column containing 10 people, how many different arrangements are there?

(3) If boys and girls are each in separate columns, and the internal order within each column does not matter, only considering who pairs with whom, how many different pairings are possible?

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