
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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