
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 following property: 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 2002 is the largest element of  $\mathcal{S},$  what is the greatet number of elements that  $\mathcal{S}$  can have?

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