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That screen indicates that the firmware/software of your iPhone is corrupted. Your only option is to force-restore the iPhone to factory settings through iTunes application (from computer) by putting it to DFU (Device Firmware Upgrade) mode, which will erase everything from it.
But if you've backed up your iPhone to iTunes on your computer or iCloud, then you can restore all your data back after restoring your iPhone from iTunes/iCloud. You can also redownload all your past purchases for free using the same account.
Follow the steps below to restore your iPhone by putting it to DFU Mode.
First download and install latest iTunes on your computer from here : https://www.apple.com/itunes/download/
Then connect the iPhone to your computer with the USB Cable.
Open iTunes application on your computer.
On your iPhone press and hold the Home button and the Power button at the same time.
After EXACTLY 8 seconds release the Power button. Continue holding the home button until your iTunes shows a message telling you that it has detected an iPhone in recovery mode.
Click OK or Restore
Note : Put it to DFU Mode while connected to computer and also in DFU Mode your iPhone will remain blank and will be unresponsive. It is normal. If it is not showing blank/black screen, then repeat the steps again. Timing is critical when you put it to DFU Mode. So keep trying until it goes to DFU Mode. And it will take a while to restore it.
Remember, it may take a while to restore the iPhone (the iOS firmware has to be downloaded, you will see the Download button - a downward arrow button with spinning circle around it, at the top right corner of iTunes screen. Click it to see the time remaining)
Please let me know if that didn't work, so I will continue to help you until we fix the problem.