
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 $E(1,0)$, $K(-1,0)$, and a moving point $P$ on the plane, satisfying $|\overrightarrow{PE}|\cdot|\overrightarrow{KE}|=\overrightarrow{PK}\cdot\overrightarrow{EK}$.

(1) Find the equation of the trajectory $C$ of point $P$;  
(2) A line $l$ passing through point $K$ intersects $C$ at points $A$ and $B$ (point $A$ is above the x-axis), the symmetric point of $A$ with respect to the x-axis is $D$, and $\overrightarrow{EA}\cdot\overrightarrow{EB}=-8$, find the equation of the circumcircle of $\triangle ABD$.

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