
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]


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A circle with a radius of $1$ is divided into three sectors with area ratios of $1:2:3$. These sectors are used as the lateral surfaces of three cones. Let the radii of the bases of these three cones be $r\_1$, $r\_2$, and $r\_3$. Calculate the value of $r\_1+r\_2+r\_3$.

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