
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
In a boarding school, there are several students living in a dormitory, one of whom is the dorm leader. During New Year's Day, each student in the dormitory exchanges greeting cards with each other, and each person also gives a greeting card to every dormitory administrator. Each administrator also gives a greeting card back to the dorm leader. In total, 51 greeting cards were used. How many students are living in this dormitory?

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