
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
Provide a circle in the coordinate system, whose circumference has

a) exactly 0, b) exactly 1, c) exactly 2, d) exactly 3 rational points.

(A point is rational if both of its coordinates are rational numbers.)

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