reach 是行動上你達到或是彼如手碰到天花板之類ㄉ~ for example, my hand reached the ceiling. achieve 是你達到某成績/成就~ 抽像ㄉ~ for example, I achieved my quota this month.
I don't quite understand why you want to place an "adj" after "to". If you really want to do so, you can say" Achieve to be pure.
he came so close to achieving his ambition should have been a dire warning to the other Elector Counts 他的功敗垂成，對於其他有投票權的伯爵而言，實為一大警訊。