
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 $n \times n$ chessboard, write the number 1 in the first cell of the first row, the numbers $2, 3, \ldots, n$ in the rest of the cells of the first row and the first column. In every other cell, write the number that is one less than the sum of the number at the beginning of its row and the number at the beginning of its column. Each cell is then occupied by a point mass, placed in its center, with weight equal to the number written in the cell. Where is the center of mass of this system of points? (The chessboard itself is considered weightless.)

Where is the center of mass if, instead of the reduced sum, the cells within the table are filled with the product of the two numbers?



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