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My MacBook 15 late 2013 run out of battery and now when I plug the charger it shows only charging screen and not turning on (it took more than 2 Hrs now).
Given the age of your system it’s safe to say the battery has offered you all it had and you need a new one! MacBook Pro 15" Retina (Late 2013-Mid 2014) Battery and here’s the guide to put it in MacBook Pro 15" Retina Display Late 2013 Battery Replacement