
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 values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Additionally, each integer in $\mathcal{S}$, when increased by 2, is divisible by $n$. Given that 1 and 2017 are elements of $\mathcal{S}$, what is the greatest number of elements that $\mathcal{S}$ can have?

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