
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 fixed point $C(2,0)$ and a line $l: x=8$ on a plane, $P$ is a moving point on the plane, $PQ \perp l$, with the foot of the perpendicular being $Q$, and $\left( \overrightarrow{PC}+\frac{1}{2}\overrightarrow{PQ} \right)\cdot \left( \overrightarrow{PC}-\frac{1}{2}\overrightarrow{PQ} \right)=0$.

(1) Find the trajectory equation of the moving point $P$;

(2) If $EF$ is any diameter of circle $N: x^{2}+(y-1)^{2}=1$, find the maximum and minimum values of $\overrightarrow{PE}\cdot \overrightarrow{PF}$.

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