
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
Let  $n$  be a positive integer. For each  $4n$ -tuple of nonnegative real numbers  $a_1,\ldots,a_{2n}$ ,  $b_1,\ldots,b_{2n}$  that satisfy  $\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n$ , define the sets
\[A:=\left\{\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:i\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\},\]
\[B:=\left\{\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:j\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\}.\]
Let  $m$  be the minimum element of  $A\cup B$ . Determine the maximum value of  $m$  among those derived from all such  $4n$ -tuples  $a_1,\ldots,a_{2n},b_1,\ldots,b_{2n}$ .

*Proposed by usjl.*

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