
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 the line $l_1: 3x+4y-5=0$, and the circle $O: x^2+y^2=4$.

(1) Find the length of the chord intercepted by circle $O$ on line $l_1$;  
(2) If the line $l_2$ passing through the point $(-1, 2)$ is perpendicular to $l_1$, and $l_2$ is tangent to a circle $M$ whose center lies on the line $x-2y=0$, and circle $M$ is divided into two arcs by line $l_1$ with the arc length ratio of 2:1, find the equation of circle $M$.

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