
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 $PQR$ has side lengths $PQ=5$, $QR=6$, and $RP=7$. Points $S$ and $T$ are on ray $PQ$ with $PQ<PS<PT$. The point $U \neq R$ is a point of intersection of the circumcircles of $\triangle PRS$ and $\triangle QRT$ satisfying $SU=3$ and $TU=8$. Find $PT$, which can be expressed as $\tfrac{m+n\sqrt{p}}{q}$, where $m$, $n$, $p$, and $q$ are positive integers such that $m$ and $q$ are relatively prime, and $p$ is not divisible by the square of any prime.

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