
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 $PQ=9, QR=10,$ 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 $P.$ Determine $PS=\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers, and find $a+b.$

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