
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 sphere is the set of points at a fixed positive distance \( r \) from its center. Let \(\mathcal{S}\) be a set of 2010-dimensional spheres. Suppose that the number of points lying on every element of \(\mathcal{S}\) is a finite number \( n \). Find the maximum possible value of \( n \).

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