Late 2011 Macbook pro"s status light comes on and goes off when press the start button,
status light comes on when press the start button but goes off, MacBook pro can not be turned on, sometime fan stays on.
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You may have temporarily put it in sleep mode.
Try resetting the SMC.
https://support.apple.com/en-us/HT201295
If that doesn't work a certified tech may need to see if the On/Off button isn't behaving right. It is better to shut down from the Apple menu.