
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
In an $n \times n$ grid, fill it with $n^{2}$ distinct real numbers $a_{i j}(i, j=1,2, \cdots, n)$. From each row, take the maximum number to get $n$ numbers, and let $x$ be the smallest of these $n$ numbers. From each column, take the minimum number to get $n$ numbers, and let $y$ be the largest of these $n$ numbers. Compare the values of $x$ and $y$.

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