While the camera is powered on, if there is no obvious damage or obstruction that would prevent the buttons from functioning properly, then the external button may be broken and need to be replaced.
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If the battery is fully charged and the camera won’t turn on then the motherboard may need to be replaced.
The external button may be broken and need to be replaced.
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While the camera is powered on, if there is no obvious damage or obstruction that would prevent the buttons from functioning properly, then the external button may be broken and need to be replaced.