
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 points $E(-2,0)$ and $F(2,0)$, the point $M$ on curve $C$ satisfies $\vec{ME} \cdot \vec{MF}=-3$. Given the fixed point $A(2,1)$ and a point $P(a,b)$ outside curve $C$ such that the tangent line from $P$ to curve $C$ touches it at $Q$, and it is given that $|PQ|=|PA|$.

(1) Find the standard equation of the circle $C$.
(2) Find the minimum value of the length of the segment $|PQ|$.

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