
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
A  $100 \times100$  chessboard has a non-negative real number in each of its cells. A chessboard is **balanced** if and only if the numbers sum up to one for each column of cells as well as each row of cells. Find the largest positive real number  $x$  so that, for any balanced chessboard, we can find  $100$  cells of it so that these cells all have number
greater or equal to  $x$ , and no two of these cells are on the same column or row.

*Proposed by CSJL.*

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