
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  $n \geq 2$  be an integer. Lucia chooses  $n$  real numbers  $x_1,x_2,\ldots,x_n$  such that  $\left| x_i-x_j \right|\geq 1$  for all  $i\neq j$ . Then, in each cell of an  $n \times n$  grid, she writes one of these numbers, in such a way that no number is repeated in the same row or column. Finally, for each cell, she calculates the absolute value of the difference between the number in the cell and the number in the first cell of its same row. Determine the smallest value that the sum of the  $n^2$  numbers that Lucia calculated can take.

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