
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 $10$ inches, length $14$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. The centers of the three faces meeting at a corner form a triangle with an area of $24$ square inches. Determine $m+n$.

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