
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  $A$  be a set with  $2n$  elements, and let  $A_1, A_2...,A_m$  be subsets of  $A$ e ach one with n elements. Find the greatest possible m, such that it is possible to select these  $m$  subsets in such a way that the intersection of any 3 of them has at most one element.

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