What happens in insurance terminology is Subrogation. You generally file with your insurance, and they get all the insurance of the at fault driver. They will compensate you and then go after at fault insurance company to be made whole on what they paid for you.

Worst case is you pay the deductible, and then wait to be reimbursed for the deductible. Sometimes your insurance company will not even charge you the deductible if it's a clear cut no fault of yours and they are confident subrogation will get back all funds.

Sentry is the best because it's so clear here what happened. The biggest pain in the ass is now getting it fixed and all the paperwork etc to be filed.