
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
In a group of  $n > 20$  people, there are some (at least one, and possibly all) pairs of people that know each other. Knowing is symmetric; if Alice knows Blaine, then Blaine also knows Alice. For some values of  $n$  and  $k,$  this group has a peculiar property: If any  $20$  people are removed from the group, the number of pairs of people that know each other is
at most  $\frac{n-k}{n}$  times that of the original group of people.
(a) If  $k = 41,$  for what positive integers  $n$  could such a group exist?
(b) If  $k = 39,$  for what positive integers  $n$  could such a group exist?

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