
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
The numbers from 1 to  $n^2$   are randomly arranged in the cells of a  $n \times n$  square ( $n \geq 2$ ). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated.  Let us call the **characteristic** of the arrangement the smallest of these  $n^2\left(n-1\right)$  fractions. What is the highest possible value of the characteristic ?

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