
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 $PR = 9,$ $QR = 40,$ and the right angle at $R.$ Point $S$ is the midpoint of $PQ,$ and $T$ is on the same side of line $PQ$ as $R$ such that $PT = QT = 41.$ Given that the area of triangle $RTS$ 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}
user