
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 circle ⊙O with equation $x^2 + y^2 = 1$ and a fixed point $A(2, 1)$, a tangent line PQ is drawn from a point $P(a, b)$ outside ⊙O to ⊙O with Q as the point of tangency, satisfying $|PQ| = |PA|$.
1. Find the relationship between the real numbers $a$ and $b$.
2. Find the minimum length of line segment PQ.
3. If a circle ⊙P with center P intersects with ⊙O, determine the 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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