
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
For any positive integer \(k\), find the smallest positive integer \(f(k)\) such that there exist 5 sets \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}\) satisfying the following conditions:
(1) \(|S_{i}| = k\) for \(i=1,2,3,4,5\);
(2) \(S_{i} \cap S_{i+1} = \varnothing \text{ (with } S_{6} = S_{1}\text{)}, i=1,2,3,4,5\);
(3) \(\left|\bigcup_{i=1}^{5} S_{i}\right| = f(k)\).

Also, what is the result when the number of sets is a positive integer \(n (n \geq 3)\)?

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