
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 \( p, q, r \) be real numbers and let \( A, B, C \) be points such that the midpoint of \( \overline{BC} \) is \( (p,0,0) \), the midpoint of \( \overline{AC} \) is \( (0,q,0) \), and the midpoint of \( \overline{AB} \) is \( (0,0,r) \). Furthermore, let \( p = l+1, q = m+1, r = n+1 \) where \( l, m, n \) are initially given constants. Find
\[
\frac{AB^2 + AC^2 + BC^2}{(l+1)^2 + (m+1)^2 + (n+1)^2}.
\]

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