
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 regular tetrahedron has a side length of \(2\). A plane parallel to one of its faces cuts through the tetrahedron at a height of \(1\) from the base, forming a smaller, similar tetrahedron and a truncated tetrahedron. Calculate the area of the triangle formed by the intersection of the plane and the tetrahedron. The area can be expressed in the form \( \frac{a\sqrt{b}}{c} \), where \(a\), \(b\), and \(c\) are positive integers with \(a\) and \(c\) relatively prime, and \(b\) not divisible by the square of any prime. What is \(a+b+c\)?
A) 8
B) 9
C) 10
D) 11
E) 12

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