
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
In triangle $PQR$, the sides are such that $PQ = 5$, $QR = 6$, and $PR = 7$. Points $S$ and $T$ are on ray $PQ$ such that $PQ < PS < PT$. A point $U \neq R$ is on the intersection of the circumcircles of $\triangle PRS$ and $\triangle QRT$ such that $SU = 3$ and $TU = 8$. Determine the length of $PT$ which can be expressed as $\frac{e+f\sqrt{g}}{h}$, where $e$, $f$, $g$, and $h$ are positive integers such that $e$ and $h$ are relatively prime, and $g$ is not divisible by the square of any prime. Find $e+f+g+h$.

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