
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 an island, there are 100 knights and 100 liars, each of whom has at least one friend. Knights always tell the truth, while liars always lie. One morning, each resident said either the phrase "All my friends are knights" or "All my friends are liars," with each phrase being stated by exactly 100 people. Find the smallest possible number of pairs of friends where one is a knight and the other is a liar.

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