
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
In a cube $ABCDEFGH$, the coordinates of vertices are set in a conventional cube alignment with $A(0, 0, 0)$, $B(2, 0, 0)$, $C(2, 0, 2)$, $D(0, 0, 2)$, $E(0, 2, 0)$, $F(2, 2, 0)$, $G(2, 2, 2)$, and $H(0, 2, 2)$. Let $M$ and $N$ be the midpoints of the segments $\overline{EB}$ and $\overline{HD}$, respectively. Determine the ratio $S^2$ where $S$ is the ratio of the area of triangle $MNC$ to the total surface area of the cube.
A) $\frac{1}{144}$
B) $\frac{17}{2304}$
C) $\frac{1}{48}$
D) $\frac{1}{96}$

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