
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
For every integer \( n \geqslant 1 \), consider the \( n \times n \) table with entry \(\left\lfloor\frac{ij}{n+1}\right\rfloor\) at the intersection of row \( i \) and column \( j \), for every \( i=1, \ldots, n \) and \( j=1, \ldots, n \). Determine all integers \( n \geqslant 1 \) for which the sum of the \( n^2 \) entries in the table is equal to \(\frac{1}{4} n^2(n-1)\).

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