
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  $A=(20,100)$  and  $D=(21,107).$   The trapezoid has no horizontal or vertical sides, and  $\overline{AB}$  and  $\overline{CD}$  are the only parallel sides.  The sum o f the absolute values of all possible slopes for  $\overline{AB}$  is  $m/n,$  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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