
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
At a conference there are  $n$  mathematicians. Each of them knows exactly  $k$  fellow mathematicians. Find the smallest value of  $k$  such that there are at least three mathematicians that are acquainted each with the other two.

<span style="color:#BF0000">Rewording of the last line for clarification:</span>
Find the smallest value of  $k$  such that there (always) exists  $3$  mathematicians  $X,Y,Z$  such that  $X$  and  $Y$  know each other,  $X$  and  $Z$  know each other and  $Y$  and  $Z$  know each other.

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