
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
Given coordinates of rectangle $PQRS$ are $P=(5,15)$ and $R=(6,23)$. The rectangle has no vertical or horizontal sides. If all other vertices $(Q, S)$ have integer coordinates, find the sum of the absolute values of all possible slopes for $\overline{PQ}$ where each slope is simplified to its lowest form $m/n$ and $m, n$ are relatively prime integers. Return the sum $m+n$.

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