Unable to apply math strategy for this question

GRE Verbal Problem Solving
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How can math strategy be applied to this question? Are there other strategies I could use to solve this question easily?

Thanks!

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I think you are wrong. For blank i, one first just knows that there are two possibilities: either it can bolster the assertion, or it can hamper the assertion. Now, if you choose option one (bolster), then in blank ii you have to choose something that says that the the two movements were closely linked together (service of BOTH woman suffrage AND theā€¦). Conversely, if you choose option two (hamper), then you have to look for a word that says that the two movements were intrinsically opposed. Looking at the answer choices, for Blank ii just E makes sense, ergo blank i has to be A.

Though I havenā€™t checked the answer keyā€¦ Does anybody know if this is correct?

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It seems I did not understand the word inimical correctly, rather, did not use it correctly.

If we were to use inimical, we should use undermine. So yeah you are right.

Looking by other forums, I think the answer is Undermines and Inimical

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I marked the correct answers with a red tick

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I think the key is the first blank. In the first blank thereā€™s no way to know if they helped or hurt the assertion. But we do have those 2 possibilities my guesses were the words ā€˜supportā€™ and ā€˜hurtā€™ as the double possibility for blank one.
Then thereā€™s the phrase energetic work in the service of that means they worked for the movement. That implies that then entire sentence has to be a net positive.
Then for the second blank, i guessed helpful and harmful. But could not narrow it down further.
Then i went to the answer choices, diffuse is off topic so eliminate it. Then in the first black, support and underscore basically mean the same thing so they canā€™t be the answer. That means undermine is the answer.
If the assertion is undermined then that means the second blank must have been a bad thing. That eliminates predominant too. Plus, inimical literally means intended to cause harm.
Hope that makes sense.

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This bad boy is a double possibility. The two sentences support each other, but there is a blank in each idea, leading to two possibilities.

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In general, do ā€œthe two movements were diffuse/predominantā€ make sense? I ask this because if I can eliminate those two options for the second blank, I can fill in the first blank too.