
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 $C\_1$: $x^{2}+y^{2}=4$, $C\_2$: $x^{2}+y^{2}-2x-4y+4=0$, and line $l$: $x+2y=0$, find the equation of the circle that passes through the intersection points of the two circles and is tangent to line $l$.

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