Analysis of an Argument discussion: What's your strategy?

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I don't think that my current method works well. I tend to weaken each hole in the logic that leads to the conclusion, one flaw at a time, holding the rest of the statement constant. In other words, I usually discredit each assumption individually, to show that each assumption cannot be upheld even if the other assumptions can be taken as true. Does anyone, especially an instructor or someone who does well on AWA's have another strategy?