
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
$ABCD$ is a rectangular sheet of paper. Points $E$ and $F$ are on $AB$ and $CD$ respectively such that $BE < CF$. If $BCFE$ is folded over $EF$, $C$ maps to $C'$ on $AD$ and $B$ maps to $B'$ such that $\angle{AB'C'} \cong \angle{B'EA}$. If $AB' = 7$ and $BE = 17$, then the area of $ABCD$ can be expressed as $p + q\sqrt{r}$ square units, where $p, q,$ and $r$ are integers and $r$ is not divisible by the square of any prime. Compute $p + q + r$.

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