
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
Consider the set \( S = \{1, 2, \ldots, 30\} \). There are 10 subsets, \( A_1, A_2, \ldots, A_{10} \), each containing 3 elements, where any two subsets \( A_i \) and \( A_j \) have at least one element in common. Define \( n_i \) as the number of subsets \( A_j \) to which element \( i \) belongs. Determine the smallest possible value of \( \max (n_i) \).

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