
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
It is known that the center of circle C is on the line $x-2y=0$.
(1) If circle C is tangent to the positive half of the y-axis, and the chord obtained by intersecting 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;
(3) Given point N(0, 3), the radius of circle C is 3, and the center C is in the first quadrant. If there exists a point M on circle C such that $MN=2MO$ (O is the origin), find the range of values for the y-coordinate of the center C.

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