
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
On Facebook, there is a group of people that satisfies the following two properties: 
(i) there exists a positive integer \( k \) such that any subset of \( 2k-1 \) people in the group contains a subset of \( k \) people in the group who are all friends with each other, and 
(ii) every member of the group has 2011 friends or fewer.

(a) If \( k=2 \), determine, with proof, the maximum number of people the group may contain.
(b) If \( k=776 \), determine, with proof, the maximum number of people the group may contain.

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