
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.) For all positive integer  $k$  find the smallest positive integer  $f(k)$  such that  $5$  sets  $s_1,s_2, \ldots , s_5$  exist satisfying:
 **i.** each has  $k$  elements;**ii.**  $s_i$  and  $s_{i+1}$  are disjoint for  $i=1,2,...,5$  ( $s_6=s_1$ )**iii.** the union of the  $5$  sets has exactly  $f(k)$  elements.

b.) Generalisation: Consider  $n \geq 3$  sets instead of  $5$ .

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