
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 with a side length of 8 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 13, 15, and 17 above the plane. Find the distance from vertex $A$ to the plane, which can be expressed as $\frac{p - \sqrt{q}}{r}$, where $p$, $q$, and $r$ are positive integers and $p+q+r < 2000$.

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