
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 C with an inscribed rectangle, the coordinates of two vertices on one of its diagonals are P(1, -2) and Q(3, 4).  
(1) Find the equation of circle C;  
(2) If the chord intercepted by the line $y=2x+b$ on circle C has a length of $2\sqrt{5}$, find the value of $b$.

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