
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
At a certain school, the ratio of boys to girls is  $1:3$ .  Suppose that: $\bullet$  Every boy has most  $2013$  distinct girlfriends. $\bullet$  Every girl has at least  $n$  boyfriends. $\bullet$  Friendship is mutual.

Compute the largest possible value of  $n$ .

*Proposed by Evan Chen*

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