
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
We glue together rigid sheets to form a paper model of a regular octahedron with side length $10 \text{ cm}$, then remove two adjacent faces. As a result, the remaining faces become movable along the edges. Determine the distance between the two endpoints of the removed edge in every possible position. What range of numbers do these distances fall within?

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