
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 that the center of circle $C$ lies on the line $x-2y=0$.

1. If circle $C$ is tangent to the positive semi-axis of the $y$-axis, and the chord intercepted by the circle on the $x$-axis has a length of $2 \sqrt {3}$, find the standard equation of circle $C$.
2. Under the conditions of question 1, line $l$: $y=-2x+b$ intersects circle $C$ at two points $A$ and $B$. If a circle passing through the origin $O$ has $AB$ as its diameter, find the value of the real number $b$.

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