
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 whose center is on the line $x-2y=0$.
(1) If circle C is tangent to the positive half of the y-axis, and the chord it cuts on the x-axis is $2\sqrt{3}$ long, find the standard equation of circle C;
(2) Under the condition of (1), the line $l: y=-2x+b$ intersects circle C at two points A and B. If the circle with AB as its diameter passes through the origin O, find the value of the real number b.

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