
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
The equation \( ax^{2} - bx + c = 0 \) has two distinct roots \( x_{1}, x_{2} \). What is the maximum number of elements in the set \( \{a, b, c, x_{1}, x_{2}\} \) that can be prime (and hence natural) numbers? If a prime number appears twice in the set, it must be counted twice. Note that the number 1 is not considered a prime number.

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