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Try using your phone until this happens again. Then charge the phone back up to 100% without trying to power it on and see if the behaviour remains.
Like @Phy suggested, try a software repair using PC Companion.
In PC Companion you press start on "Support Zone" > Start on "Update the phone/tablet software" > "Repair phone/tablet" (the blue clickable link in the message) > Follow these steps without having your device connected to the PC.
Follow the instruction that appears in the program and do not connect your device until this guide tells you how to connect it.
The software repair will remove all data stored in the phone so don't forget to make a backup in PC Companion before you try the software repair. You can do a backup using the Backup & Restore option in PC Companion.
You can post the pictures here (screenshots) or you could upload them to box or dropbox and post the link here.