
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 received the following statements about a tetrahedron and its circumscribed sphere:

I. One base edge is equal in length to the diameter of the sphere.

II. The ratio of the other two base edges is $3:4$.

III. The volume of the tetrahedron is $40 \mathrm{~cm}^{3}$.

IV. The edges of the tetrahedron are equal.

V. The base of the tetrahedron is a right triangle.

VI. The height of the tetrahedron is equal to the radius of the sphere.

Additionally, it is given that one of the six statements is false, while the others are true.

What are the lengths of the edges of the tetrahedron?

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