
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 the circle $O$: $x^{2}+y^{2}=1$ and the point $M(1,4)$.
(1) Draw a tangent line through point $M$ to the circle $O$, and find the equation of the tangent line;
(2) Find the equation of the circle $M$ with center at point $M$ and whose chord length intercepted by the line $y=2x-8$ is $8$;
(3) Let $P$ be any point on the circle $M$ found in part (2). Draw a tangent line through point $P$ to the circle $O$, and let $Q$ be the point of tangency. Investigate whether there exists a fixed point $R$ in the plane such that $\frac{PQ}{PR}$ is constant. If such a point exists, find its coordinates and the corresponding constant value. If not, explain why.

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