
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 a finite set  $E$  of cardinality  $n \geq 3$ , let  $f(n)$  denote the maximum number of  $3$ -element subsets of  $E$ , any two of them having exactly one common element. Calculate  $f(n)$ .

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