
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
Let \( R = (10, 8) \). The lines whose equations are \( 7y = 24x \) and \( 13y = 5x \) contain points \( P \) and \( Q \), respectively, such that \( R \) is the midpoint of \( \overline{PQ} \). Find the length of \( PQ \), expressed in simplest form as \( \frac{a}{b} \), where \( a \) and \( b \) are relatively prime positive integers. Then, find \( a + b \).

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