
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 closed rectangular box with a square base has dimensions with width \( w \) and height \( h \), and an extra vertical extension (like a chimney) standing on the center of the top. This extension is a smaller square prism of height \( d \) and the same width \( w \). This box including its extension is to be wrapped with a square sheet of wrapping paper. The combined structure is centered on the wrapping paper with vertices of the base as well as the top of the extension laying on the midlines of the square sheet of paper. The four corners of the wrapping paper are folded up to meet at the center top of the extension. What is the area of the wrapping paper required?
**A)** \( w^2 + 2wh + 2wd \)  
**B)** \( (w\sqrt{2} + h + d)^2 \)  
**C)** \( 2w^2 + 2wh + 2wd \)  
**D)** \( 4(w+h+d)^2 \)

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