
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), F(2, 0), a moving point M on curve C satisfies $$\overrightarrow{ME} \cdot \overrightarrow{MF} = -3$$, and a fixed point A(2, 1). From a point P(a, b) outside curve C, a tangent line PQ is drawn to curve C at point Q, satisfying $|PQ| = |PA|$.  
(1) Find the equation of curve C;  
(2) Find the minimum length of segment PQ;  
(3) If a circle P with center at P intersects curve C at common points, find the standard equation of circle P when its radius is at its minimum value.

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