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im cam, a licenced diagnostic tech.
these radios from 05 to 07 are very well known for this turning off and on again.
this is usually an internal problem with the radio's itself.
i would remove the radio and make sure the connectors are properly seated and wiggle them while the radio is turned on and see if it is cutting out while doing this.
if it does not cut out when doing this than you will be looking at in internal radio problem and will require replacing .
let me know how your repair gos!
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