
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
Consider a  $ n\times n $  square grid which is divided into  $ n^2 $  unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an  $n$ -staircase. Find the number of ways in which an  $n$ -stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.

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