
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 school there are  $1200$  students. Each student is part of exactly  $k$  clubs. For any  $23$  students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of  $k$ .

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