
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 $R = (10, 4)$. The lines whose equations are $5y = 12x$ and $7y = 2x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\overline{PQ}$. Calculate the length of $PQ$ and express the result in the form $\frac{m}{n}$, where $m$ and $n$ are coprime integers. Find $m + n$.

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