
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 two circles $C_1: x^2+y^2=4$ and $C_2: x^2+y^2-2x-4y+4=0$, and the line $l: x+2y=0$, find the equation of the circle that passes through the intersection points of $C_1$ and $C_2$ and is tangent to the line $l$.

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