
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 octahedron has side length $2$. A plane at an angle of $45^\circ$ to two of its opposite faces cuts the octahedron into two solids. The polygon formed by the intersection of the plane and the octahedron has an area of $\frac{a\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, with $a$ and $c$ relatively prime, and $b$ not divisible by the square of any prime. Find $a + b + c$.
A) 10  
B) 11  
C) 12  
D) 13  
E) 14

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