
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 sides \( PQ=26 \), \( PR=28 \), and \( QR=30 \). Points \( S \) and \( T \) lie on \( \overline{PQ} \) and \( \overline{PR} \), respectively, such that \( \overline{ST} \) is parallel to \( \overline{QR} \) and contains the center of the inscribed circle of triangle \( PQR \). Find \( ST \) as a fraction \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. What is \( m+n \)?

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