
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 side lengths are $PQ=9$, $QR=10$, and $RP=11$. Circle $\omega_1$ passes through $Q$ and is tangent to line $RP$ at $P$. Circle $\omega_2$ passes through $R$ and is tangent to line $PQ$ at $P$. Let $S$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $P$. Find the length of $PS$ in simplest form, and express your result as $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers.

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