
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 pentagon $ABCDE,$ side $\overline{AB}$ is extended past $B$ to $A'$ such that $A'B = 2AB.$ Similarly, extend $\overline{BC}$ past $C$ to $B'$ so that $B'C = 2BC$, $\overline{CD}$ past $D$ to $C'$ so that $C'D = 2CD$, $\overline{DE}$ past $E$ to $D'$ so that $D'E = 2DE$, and finally extend $\overline{EA}$ past $A$ to $E'$ so that $E'A = 2EA.$ After these extensions, the original points $A, B, C, D, E$ are erased. You are left with the points $A', B', C', D', E'$ and need to reconstruct the original pentagon $ABCDE.$

Determine if there exist real numbers $p, q, r, s, t$ such that:
\[
\overrightarrow{A} = p \overrightarrow{A'} + q \overrightarrow{B'} + r \overrightarrow{C'} + s \overrightarrow{D'} + t \overrightarrow{E'}.
\]

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