
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
Given a triangle $\triangle ABC$ with vertices at $A(-1,0)$, $B(1,0)$, and $C(3,2)$, and its circumcircle $H$, for any point $P$ on the line segment $BH$, if there exist two distinct points $M$ and $N$ on the circle with center $C$ such that $M$ is the midpoint of the line segment $PN$, find the range of values for the radius $r$ of circle $C$.

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