
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
On an $11 \times 11$ grid, 22 cells are marked such that exactly two cells are marked in each row and each column. If one arrangement of the marked cells can be transformed into another by swapping rows or columns any number of times, the two arrangements are considered equivalent. How many distinct (nonequivalent) arrangements of the marked cells are there?

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