
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
\(S\) is a set of complex numbers such that if \(u, v \in S\), then \(uv \in S\) and \(u^{2} + v^{2} \in S\). Suppose that the number \(N\) of elements of \(S\) with absolute value at most 1 is finite. What is the largest possible value of \(N\)?

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