
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
We call two people a friendly couple if they know each other, and we call them a non-friendly couple if they do not know each other (being friends or not being friends is always mutual). Let \(m, n\) be natural numbers. Find the smallest natural number \(k\) such that the following holds: In every group of \(k\) people, there are always \(2m\) people who form \(m\) disjoint friendly pairs, or there are \(2n\) people who form \(n\) disjoint non-friendly pairs.

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