
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
Let \( S \) be a finite set of integers. Suppose that for any two distinct elements \( p \) and \( q \) in \( S \), there exist three elements \( a \), \( b \), and \( c \) in \( S \) (where \( a \), \( b \), and \( c \) do not need to be distinct, and \( a \neq 0 \)), such that the polynomial \( F(x) = ax^2 + bx + c \) satisfies \( F(p) = F(q) = 0 \). Determine the maximum number of elements in \( S \).

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