
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 cube in 3-dimensional space has eight vertices at $(\pm 1, \pm 1, \pm 1)$. Each edge of the cube is halved, and midpoints of each edge are joined to form smaller cubes inside the main one. Find the ratio of the volume of one smaller internal cube to the original cube. Express this ratio in simplest form as $m/n$ and find $m + n$.

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