
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
We call a set of professors and committees on which they serve a university if
1. Given two distinct professors, there is one and only one committee on which they both serve.
2. Given any committee \( C \) and any professor \( P \) not on that committee, there is exactly one committee on which \( P \) serves and no professors on committee \( C \) serve.
3. There are at least two professors on each committee, and there are at least two committees.

What is the smallest number of committees a university can have?

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