
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
Triangle $PQR$ is a right triangle with $PQ = 8,$ $PR = 15,$ and right angle at $R.$ Point $S$ is the midpoint of $PQ,$ and $T$ is on the same side of line $PQ$ as $R$ so that $PT = 15.$ Given that the area of triangle $TRS$ may be expressed as $\frac{x \sqrt{y}}{z},$ where $x,$ $y,$ and $z$ are positive integers, $x$ and $z$ are relatively prime, and $y$ 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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