
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
Consider a square pyramid with a square base. The vertices of the pyramid are at coordinates $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$, and $(0,0,0,0)$ in 4-space. The center of the square base is one of the vertices of a smaller square pyramid, and the other pyramid's vertices are the midpoints of the edges connecting the apex to the base. Find the ratio of the volume of the smaller pyramid to that of the larger pyramid, and calculate $m+n$ if this ratio is given by $m/n$, where $m$ and $n$ are relatively prime positive integers.

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