
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 $C_1: {(x-1)^2}+{(y-2)^2}=4$ and $C_2: {(x-2)^2}+{(y-1)^2}=2$ intersect at points $A$ and $B$. Line $l$ is parallel to line $AB$, tangent to circle $C_{2}$, and intersects circles $C_{1}$ and $C_{2}$ at points $M$ and $N$ respectively. Find the length of $|MN|$.

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