
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
Given a room containing $N$ people, $N > 4$, exactly two persons have not shaken hands with everyone else in the room. Each of these two persons has not shaken hands with two specific individuals. Determine the maximum number of people in the room that could have shaken hands with everyone else.

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