
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 quadrilateral $PQRS$, side $\overline{PQ}$ is extended past $Q$ to $P'$ so that $P'Q = 2PQ$. Likewise, extend $\overline{QR}$ to $Q'$ such that $QR' = QR$, $\overline{RS}$ to $R'$ with $SR' = SR$, and $\overline{SP}$ to $S'$ such that $PS' = 3PS$.

After extending these sides, the points $P$, $Q$, $R$, and $S$ are removed. Using only the locations of points $P'$, $Q'$, $R'$, and $S'$, reconstruct the quadrilateral $PQRS$. Assume there exist real numbers $x$, $y$, $z$, and $w$ such that:
\[
\overrightarrow{P} = x\overrightarrow{P'} + y\overrightarrow{Q'} + z\overrightarrow{R'} + w\overrightarrow{S'}.
\]
Enter the ordered quadruple $(x,y,z,w)$.

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