
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
The coordinates of the vertices of isosceles trapezoid \(ABCD\) are all integers, with \(A=(30,200)\) and \(D=(31,217)\). The trapezoid has no horizontal or vertical sides, and \(\overline{AB}\) and \(\overline{CD}\) are the only parallel sides. Calculate the sum of the absolute values of all possible slopes for \(\overline{AB}\), expressed in simplest form as \(m/n\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).

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