Why Would Serena Give Jack the Cups?

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I think Syrena has the ability to see into men's hearts hence why she knew Phillip was "different' from the other sailors. That being said I think she was able to see into Jack's heart and see that he didn't like Blackbeard and that he had every intention of killing him and saving Angelica. She probably also saw that deep deep deep down inside, Jack Sparrow was a good man. Also Syrena knew about the prophecy and no doubt wanted Blackbeard to meet his doom as prophesied one way or another whether it was by Barbosa's doing or Jack's. Furthermore, Syrena no doubt would've wanted revenge on Black beard for his treatment of her but was in no position to do anything about it ( she's not a warrior and is clumsy and slow on land...she can't walk when she has legs...she wouldn't be able to cause physical harm to blackbeard) so her best bet was to help Jack knowing full well that Jack didn't like Blackbeard and would most likely kill him if given the means. As a result she provided him with the means ( the chalices) and gave him an added warning not to waste her tear but to use it against blackbeard who had caused such havoc.

She tells him why when she says 'don't waste my tear'
Mermaids don't cry easily, their tears are rare and precious (at least in the POTC lore) and therefore she wanted it to be used and not just given for nothing