
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
Find the greatest constant  $\lambda$  such that for any doubly stochastic matrix of order 100, we can pick  $150$  entries such that if the other  $9850$  entries were replaced by  $0$ , the sum of entries in each row and each column is at least  $\lambda$ .

Note: A doubly stochastic matrix of order  $n$  is a  $n\times n$  matrix, all entries are nonnegative reals, and the sum of entries in each row and column is equal to 1.

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