
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
A set \( \mathcal{S} \) of distinct positive integers has the property that for every integer \( x \) in \( \mathcal{S}, \) the arithmetic mean of the set of values obtained by deleting \( x \) from \( \mathcal{S} \) is an integer. Given that 1 belongs to \( \mathcal{S} \) and that 2310 is the largest element of \( \mathcal{S}, \) and also \( n \) must be a prime, what is the greatest number of elements that \( \mathcal{S} \) can have?

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