
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 circle $O$: $x^{2}+y^{2}=1$ and point $M(4,2)$. A circle centered at point $M$ intercepts a chord of length $2 \sqrt {5}$ on the $x$-axis.
(1) Find the equation of circle $M$;
(2) Suppose $P$ is any point on circle $M$, and a tangent line is drawn from point $P$ to circle $O$, intersecting at point $Q$. Investigate whether there exists a fixed point $R$ in the plane such that $\frac{PQ}{PR}$ is a constant value. If it exists, find the point and the corresponding constant value; if not, explain the reason.

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