
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
Let $l, m, n, p$ be real numbers, and let $A, B, C$ be points such that the midpoint of $\overline{BC}$ is $(l,0,0),$ the midpoint of $\overline{AC}$ is $(0,m,0),$ and the midpoint of $\overline{AB}$ is $(0,0,n).$ Additionally, let $p$ be the z-coordinate of all vertices $A, B, C$. Find
\[
\frac{AB^2 + AC^2 + BC^2}{l^2 + m^2 + n^2 + 3p^2}.
\]

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