
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 pentagon $ABCDE$, side $\overline{AB}$ is extended past $B$ to $A'$ such that $A'B = AB$, and then $A'$ is extended past $B'$ to $A''$ such that $A''B' = A'B$. Points $B'$, $C'$, $D'$, and $E'$ are similarly constructed. After this construction, points $A$, $B$, $C$, $D$, and $E$ are erased. You are left with the points $A'$, $A''$, $B'$, $C'$, $D'$, and $E'$, and you need to reconstruct pentagon $ABCDE$.

Mathematically, there exist real numbers $p$, $q$, $r$, $s$, $t$, $u$ such that
\[
\overrightarrow{A} = p \overrightarrow{A'} + q \overrightarrow{A''} + r \overrightarrow{B'} + s \overrightarrow{C'} + t \overrightarrow{D'} + u \overrightarrow{E'}.
\]
Find the ordered tuple $(p, q, r, s, t, u)$.

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