
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
Bráulia cut a cube into many smaller cubes with edge length of $1 \text{ cm}$ by making cuts parallel to its faces. For example, if this cube had sides of $4 \text{ cm}$, the cuts would produce:

![Image of cube cuts](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-02.jpg?height=360&width=374&top_left_y=551&top_left_x=744)

However, the edge length of this cube is unknown.

a) After cutting the cube, Bráulia counted the small cubes of $1 \text{ cm}$ side, totaling 512. What was the edge length of the original cube?

b) Laura does the same with another cube, for which the edge length is also unknown. After the cut, Laura counts 512 small cubes that previously had no face in contact with the exterior of the cube. What was the edge length of this cube?

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