
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 \( \mathcal{T} \) of distinct positive integers has the property that for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 3003 is the largest element of \( \mathcal{T}, \) what is the greatest number of elements that \( \mathcal{T} \) can have?

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