
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
Three spheres of radius 2 and one sphere of radius 3 are mutually tangent, with the smaller spheres resting on a flat floor and the larger sphere resting on them. A tetrahedron circumscribes these four spheres. Determine the height of the tetrahedron from the floor to the opposite vertex.
**A)** \( \frac{10\sqrt{3}}{3} \)  
**B)** \( \frac{11\sqrt{3}}{3} \)  
**C)** \( \frac{\sqrt{177} + 9\sqrt{3}}{3} \)  
**D)** \( \frac{12\sqrt{3}}{3} \)  
**E)** \( \frac{13\sqrt{3}}{3} \)

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