
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
Let  $F$  be a family of subsets of  $S = \left \{ 1,2,...,n \right \}$  ( $n \geq 2$ ). A valid play is to choose two disjoint sets  $A$  and  $B$  from  $F$  and add  $A \cup B$  to  $F$  (without removing  $A$  and  $B$ ).

Initially,  $F$  has all the subsets that contain only one element of  $S$ . The goal is to have all subsets of  $n - 1$  elements of  $S$  in  $F$  using valid plays.

Determine the lowest number of plays required in order to achieve the goal.

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