
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 circle $C$ that passes through points $A(1,2)$ and $B(1,10)$, and is tangent to the line $x-2y-1=0$.

(1) Find the equation of circle $C$;

(2) Let $P$ be any point on circle $C$, and $Q$ be a fixed point at $(-3,-6)$. As point $P$ moves along circle $C$, find the trajectory equation of the midpoint $M$ of line segment $PQ$.

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