
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 prism has dimensions of 1 by 1 by 2. Calculate the sum of the areas of all triangles whose vertices are also vertices of this rectangular prism, and express the sum in the form $m + \sqrt{n} + \sqrt{p}$, where $m, n,$ and $p$ are integers. Find $m + n + p$.

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