How are you powering your Stick? If you're using the TV's USB port, it might be turning the power off. If you're using the wall power supply, it might be failing. Roku devices don't have an on/off function. If they have power, they are always "on", but might enter a low power mode. You might also try a different screensaver on the Roku. Perhaps something in the one you're using is triggering the video output off.
Instead of pressing the reset button in the Stick, does anything happen if you simply press a button on the Roku remote?
Exactly what model Stick do you have? I'm just wondering if it's an older one that might be experiencing some sort of internal failure.
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