
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 a rhombus $ABCD$ on a plane where $AB=BC=CD=DA=1$, points $\overrightarrow{AA^\prime}$, $\overrightarrow{BB^\prime}$, $\overrightarrow{CC^\prime}$, and $\overrightarrow{DD^\prime}$ are vertical rays in the same direction. If $AA^{\prime} = 12$, $BB^{\prime}= 20$, $CC^\prime = 12$, and $DD^\prime = 20$, find the length of $PQ$, where $P$ and $Q$ are the midpoints of $A^{\prime} C^{\prime}$ and $B^{\prime}D^{\prime}$, respectively.

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