
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 \(EFGH\), side \(\overline{EF}\) is extended past \(F\) to \(E'\) such that \(E'F = EF\). Points \(F'\), \(G'\), and \(H'\) are similarly constructed. After this construction, points \(E\), \(F\), \(G\), and \(H\) are erased. You only know the locations of points \(E'\), \(F'\), \(G'\) and \(H'\), and want to reconstruct quadrilateral \(EFGH\).

There exist real numbers \(x, y, z, w\) such that
\[
\overrightarrow{E} = x \overrightarrow{E'} + y \overrightarrow{F'} + z \overrightarrow{G'} + w \overrightarrow{H'}.
\]

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