
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 $C$ passing through points $A(1,2)$ and $B(1,10)$, with its center $O$ in the first quadrant, and 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 $Q(-3,-6)$ be a fixed point. When point $P$ moves on circle $C$, find the trajectory equation of the midpoint $M$ of segment $PQ$.

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