
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
Given the rectangle $ABCD$ in space where $AB = 2$ units and $BC = 1$ unit, the coordinates of $A$, $B$, $C$, and $D$ are respectively $A(0,0,0)$, $B(2,0,0)$, $C(2,1,0)$, and $D(0,1,0)$. Define rays $\overrightarrow{AA'}$, $\overrightarrow{BB'}$, $\overrightarrow{CC'}$, $\overrightarrow{DD'}$ extending in the positive $z$-direction from $A$, $B$, $C$, and $D$. If $AA' = 12$, $BB' = 10$, $CC' = 20$, and $DD' = 24$, find the distance $MN$ where $M$ and $N$ are the midpoints of $A'C'$ and $B'D'$.

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