Bless you for being so caring and wanting to help support your cousin. You're right: some of the reaction within your family may be very hard to take, and may be unfair and unsupportive. And she'll need someone to be her advocate very much. She's lucky to have you.
The first thing you can help her with is getting some support in terms of dealing with sexual assault overall. I don't know what part of Chicago you're in, but you can call the police station she went to with her and get a loisting of survivor supprot groups in your area. Offer to go with her. Her finding support with others who have been through similar can be really key, and she may feel better going with someone like you than alone, initially.
Make sure you let her know you DO believe her, and that you're there for her should she need any little thing; all she needs to do is ask you. That in and of itself is huge, really.
You may also just want to be ready to be a fair mediator in discussions within your family she isn't included in about this. If you can help keep things from getting nasty, or just being hidden, that's great. Seems to me there arre also likely some residual issues to deal with in your family -- like having family parties where minors are getting drunk, for instance, which is basically a recipe for disaster and dysfunction.
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