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Well, I am sorry to hear this has happened as you are in a pretty tough situation legally. Without insurance in AZ and at fault in the accident if you are sued by them they will recover this amount plus attorney's fees.
In your dispute letter, really all you can do is dispute the amount of the damages, since you said it was your daughter's friend and son's fault. In order to do so, you need to go to at least 2 or 3 local body shops and try to get them to give you an estimate based on either photographs or other documentation you have regarding the damage to the other vehicle. In your letter you would need to use those estimates to argue that the amount they are charging is excessive and you need to make a counter offer for settlement for a lower amount.
I am sorry to say this is really the only thing you can do here based on the facts you provided being so much against you (other than sue the person who took your car without permission and was driving
it). The only other thing you can do in your letter is try to argue how their driver is at least partially at fault and if you were to go to court to litigate this both you and the insurer would incur legal costs and in the interest of fairness you would offer settlement of $2300 (which is 1/2 of the amount they claim you owe). Other than engaging in this type of negotiation, your facts you have been dealt really hurt you badly here.
We cannot actually write the letter, we do not know sufficient facts and details and it would also be considered the practice of law which is forbidden by the rules of this site. However, if you follow the guidelines above that should get you going in the right direction.
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