
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. $P$ and $Q$ are points on $AB$ and $CD$ respectively such that $BP < DQ$. If $BCPQ$ is folded over $PQ$, $D$ maps to $D'$ on $AD$ and $B$ maps to $B'$ such that $\angle{AD'B'} \cong \angle{B'PA}$. If $AD' = 7$ and $BP = 27$, then the area of $ABCD$ can be expressed as $x + y\sqrt{z}$ square units, where $x, y,$ and $z$ are integers and $z$ is not divisible by the square of any prime. Find $x + y + z$.

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