
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
Let  $n{}$  be a natural number. The playing field for a "Master Sudoku" is composed of the  $n(n+1)/2$  cells located on or below the main diagonal of an  $n\times n$  square. A teacher secretly selects  $n{}$  cells of the playing field and tells his student


- the number of selected cells on each row, and
- that there is one selected cell on each column.
The teacher's selected cells form a Master Sudoku if his student can determine them with the given information. How many Master Sudokus are there?

*Proposed by T. Amdeberkhan, M. Ruby and F. Petrov*

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