Xbox with dvi and constantly turns off by itself

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I have a Xbox 360 that has a broken HDMI jack with no sound (DVI), but still allows me to play on hi-def. Another problem with it is that it constantly shuts off by itself, reverts itself to 480x640 resolution until it heats up to a point. I have yet to find out how to fix it, and was wondering if others have had this kind of problems?
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Why does my Xbox keep turning off by itself?
If your Xbox console is unexpectedly shutting down, it may be because it's not getting enough ventilation, or there may be a problem with the power supply.How do I get my Xbox out of DVI mode?
To reset this setting, go to System > Settings in the guide, select Display & sounds, Video options, and then TV Resolution.*FIX* XBOX RANDOMLY TURNS OFF! (POWER RESET METHOD)
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