
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 $2n$ people, some of whom are friends with each other. In this group, each person has exactly $k(k \geqslant 1)$ friends. Find the positive integer $k$ such that the $2n$ people can always be divided into two groups of $n$ people each, and every person in each group has at least one friend in their own group.

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