
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
A circle C satisfies the following conditions:  
① The center of circle C is on the ray $y=2x$ ($x>0$);  
② It is tangent to the x-axis;  
③ The segment intercepted by the line $y=x+2$ has a length of $\sqrt{14}$.  
(1) Find the equation of circle C;  
(2) Through a point P on the line $x+y+3=0$, draw tangents to circle C, and let the points of tangency be E and F. Find the minimum value of the area of quadrilateral PECF, and find the value of $\overrightarrow{PE} \cdot \overrightarrow{PF}$ at this time.

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