
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
Consider a cube with edge length 2 units. Calculate the sum of the areas of all triangles whose vertices are vertices of this cube. Express the sum as \(a + \sqrt{b} + \sqrt{c}\), where \(a, b,\) and \(c\) are integers, and find \(a + b + c\).

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