Every time I have had to deal with Asurion I have always received the device next day and their questionnaire about how you lost the device is very straight forward (I believe it's just type in 500 words or less a summary of what happened and then they even ask if you lost any accessories because they will even replace those). Just keep in mind that the device they send will more than likely be a refurbished device (I've had mixed results with this but their customer service has been really great working to resolve issues with replacement devices, just be sure if there are any defects in the replacement device that you call and complain in a professional manner, they will usually have another device sent out and delivered within 24 hours).
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