
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 the triangle \( ABC \), \( AB = 585 \), \( BC = 520 \), \( CA = 455 \). Let \( P \) and \( Q \) be points on the side \( BC \), and \( R \neq A \) be the intersection of the line \( AQ \) with the circumcircle \(\omega\) of the triangle \( ABC \). Suppose \( PR \) is parallel to \( AC \) and the circumcircle of the triangle \( PQR \) is tangent to \(\omega\) at \( R \). Find \( PQ \).

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