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I am very sorry to tell you but this is 100% not legitimate. No only do we hear questions just like this on the site weekly, but at no time ever would anyone be required to pay any funds up front if the person won a lottery. Rather, at all times, without exception, any money owing would come off the lottery funds and would not come from the person who won.
I suggest that you absolutely do not pay them one cent, because you will never get the money back and they are professionals who will never be caught.