
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
A box-shaped room, dimensions $6 \times 6 \times h$, has a large sphere of radius $3$ and four smaller spheres each of radius $1$ situated at the corners of the base. Each of the small spheres is tangent to three sides of the room, and the large sphere is tangent to the four smaller spheres. What is the value of $h$?
A) $2 + 4\sqrt{2}$
B) $2 + 2\sqrt{2}$
C) $4 + 3\sqrt{4}$
D) $5 + 2\sqrt{2}$
E) $3 + 2\sqrt{3}$

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