
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 quadrilateral $ABCD,$ side $\overline{AB}$ is extended past $B$ to $A''$ so that $A''$ trisects the segment $AB$ extended twice i.e., $BA'' = \frac{2}{3} \overline{BA}.$ Points $B'',$ $C'',$ and $D''$ are similarly trisected.

Assume that points $A,$ $B,$ $C,$ and $D$ are erased. You only know the locations of points $A'',$ $B'',$ $C'',$ and $D'',$ and want to reconstruct quadrilateral $ABCD.$

Determine real numbers $p,$ $q,$ $r,$ and $s$ such that
\[
\overrightarrow{A} = p\overrightarrow{A''} + q\overrightarrow{B''} + r\overrightarrow{C''} + s\overrightarrow{D''}.
\]

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