
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 regular tetrahedron with side length 1. A tetrahedron has 4 vertices. Calculate the sum of the areas of all triangles whose vertices are also vertices of this tetrahedron. Express your answer in the form \( a + b\sqrt{c} \), where \( a \), \( b \), and \( c \) are integers. Find \( a + b + c \).

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