
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 geometric construction begins with quadrilateral $ABCD$. Each side of the quadrilateral is extended such that side $\overline{AB}$ is extended past $B$ to $E$, where $BE = 2AB$; simultaneously, sides $\overline{BC}, \overline{CD},$ and $\overline{DA}$ are similarly extended to $F, G,$ and $H$, respectively, with each new segment twice the length of the original side. Points $E' = 2F - E$, $F' = 2G - F$, $G' = 2H - G$, and $H' = 2E - H$ are constructed. Points $A, B, C,$ and $D$ are erased, leaving only points $E', F', G',$ and $H'$. The task is to reconstruct point $A$ using these points and determine the weights such that $\overrightarrow{A} = p \overrightarrow{E'} + q \overrightarrow{F'} + r \overrightarrow{G'} + s \overrightarrow{H'}$.

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