
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
Let $ABCD$ be a rectangle in the $xy$-plane where $AB=2$ and $BC=1$ with coordinates $A(0,0,0)$, $B(2,0,0)$, $C(2,1,0)$, and $D(0,1,0)$. Let $\overrightarrow{AA'}$, $\overrightarrow{BB'}$, $\overrightarrow{CC'}$, and $\overrightarrow{DD'}$ be parallel rays in space on the same side of the plane determined by $ABCD$. If $AA' = 12$, $BB' = 10$, $CC' = 16$, and $DD' = 20$, and $M$ and $N$ are the midpoints of $A'C'$ and $B'D'$ respectively, find the length of $MN$.

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