
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 \( r \) be a rational number in the interval \([-1,1]\) and let \(\theta = \cos^{-1} r\). Call a subset \( S \) of the plane good if \( S \) is unchanged upon rotation by \(\theta\) around any point of \( S \) (in both clockwise and counterclockwise directions). Determine all values of \( r \) satisfying the following property: The midpoint of any two points in a good set also lies in the set.

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