
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 rectangular box has width $15$ inches, length $20$ inches, and height $h$ inches, where $h$ is given as $\frac{p}{q}$ inches, with $p$ and $q$ being relatively prime positive integers. Three faces of the box meet at a corner. The center points of those three faces form a triangle with an area of $50$ square inches. Find $p+q$.

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