
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 family of sets  $ A_1, A_2, \ldots ,A_n$  has the following properties:

**(i)** Each  $ A_i$  contains 30 elements.
**(ii)**  $ A_i \cap A_j$  contains exactly one element for all  $ i, j, 1 \leq i < j \leq n.$ 

Determine the largest possible  $ n$  if the intersection of all these sets is empty.

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