
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 circle O: $x^2+y^2=5$ and a fixed point A(4, 3), a tangent line PQ is drawn from a point P(a, b) outside circle O to circle O at the tangent point Q, satisfying $|PQ|=|PA|$

(1) Find the relationship between the real numbers a and b;  
(2) Find the minimum length of segment PQ;  
(3) If a circle P with center P intersects with circle O at some point, find the equation of circle P when its radius is minimized.

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