
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

All the cells of a \(100 \times 100\) square table are numbered in some order from 1 to 10000. Petya colors the cells according to the following rules. First, he colors \(k\) cells at his discretion. Then, on each turn, Petya can color one yet-to-be-colored cell with number \(a\) if at least one of the two conditions is met: either there is already a colored cell with a number less than \(a\) in the same row, or there is already a colored cell with a number greater than \(a\) in the same column. What is the smallest \(k\) such that, regardless of the initial numbering, Petya will be able to color all the cells of the table in several steps?

(S. Berlov)

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