
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
$PQRS$ is a rectangular sheet of paper. $T$ and $U$ are points on $PQ$ and $RS$ respectively such that $QT > PU$. The rectangle is folded over $TU$ such that $S$ maps to $S'$ on $PQ$ and $R$ maps to $R'$ such that $\angle{PR'S'} \cong \angle{R'TQ}$. If $PR' = 7$ and $QT = 29$, find the area of rectangle $PQRS$ which 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. Compute $x + y + z$.

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