
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 that in a meeting of $N$ people, where $N>4$, two specified individuals have each not shaken hands with at least one other different person in the room, calculate the maximum number of people in the room who could have shaken hands with everyone else.

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