
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 $A$ and $B$ are two distinct points on the circle $C: (x-m)^2+(y-n)^2=9$ $(m,n \in \mathbb{R})$ (where $C$ is the center of the circle), and $| \overrightarrow{CA} + \overrightarrow{CB}| = 2 \sqrt {5}$, find the value of $|AB|$.

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