
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 two points $A(-2,1)$ and $B(5,0)$, and the center of the circle $C$ is on the line $y=2x$.

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

(2) A moving line $l$: $(m+2)x+(2m+1)y-7m-8=0$ passes through a fixed point $M$, and a line $m$ with a slope of $1$ passes through the point $M$, intersecting the circle $C$ at points $P$ and $Q$. Find the length of $PQ$.

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