What you need

Directions:
1. Go to playstation.com, and find the most recent system software. The current page for this is here: [ Register or Signin to view external links. ]

Note that there are multiple system software files -- update versions and full installs. Make sure you find the latest version (the page will display the latest version number up top). Look for the full install listed under "Perform a New Installation of the System Software), not just an update file. The file should be close to 1 GB in size (some previous versions were around 800 MB, whereas update files are around 300 MB).

Download the 800+ MB (may be more) system recovery/new install software.
2. Test your USB drive with your PS4 first. Make sure it plugs snugly into one of the USB ports on your PS4 -- some drives are too fat and can't be inserted all the way. If your drive works, move on to the next step.

3. Plug your USB drive into your computer. You can use a PC or a Mac, but you have to make sure the USB flash drive is formatted as FAT or FAT32 to avoid issues. Most flash drives come preformatted and should be ready to go, but if you want to be sure note that erasing and formatting your drive only takes a few seconds and the system may return an error with other formats. On a PC, right-click the drive and select Format from the menu. On Mac, use Disk Utility and erase and reformat using MS DOS (FAT).
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3. Create a folder titled PS4 (all caps) on the flash drive. Create another folder titled UPDATE (all caps) within that PS4 folder.

4. Move the system software download into the UPDATE folder. The file name is PS4UPDATE.PUP. If the filename is different (perhaps because you have multiple instances of update files on your computer), make sure to change it back to that name before you move on. Safely eject the USB flash drive. On Mac, hit the eject button next to your drive in Finder.

5. It's time to back up your PlayStation 4 game save data. You can either back up your saves to another (or the same) USB storage device, or if you've got PlayStation Plus, to the cloud. Either way, go to the Settings, Application Saved Data Management, and then select System Storage and copy your saved data to the USB Storage Device or to the cloud.

6. Turn off your PS4. If the amber light is on, the system is still in standby mode. Hold the off button until it turns off completely. Then unplug the power cable and remove any other wires that may get in the way of working on the system.
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7. Remove the left side of the PS4 case by simply depressing it and sliding it to the side, as shown.

The entire left portion of the casing can be removed. It's not on rollers -- you can just lift it up and off once you've slid it to the left.
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8. Look for the screw head with PlayStation button symbols on it. It's a large philips/cross-style screw at the front left of the PlayStation. It holds the hard drive case in place inside the PlayStation 4. Remove this screw.
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9. Tug the default hard drive out by pulling it forward -- it should slide out freely.
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10. You now have to remove the hard drive from its casing. Don't worry, it's super-easy.
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There are four screws, two on each side. Remove them as shown, leaving the small rubber parts in place. You should have four screws plus the silver PlayStation-branded screw to keep safe.

Remove the hard drive from its case and replace it with your new hard drive. Make sure the screw holes line up on all sides and use the screw driver to put everything back to gether.

11. Re-insert the hard drive caddy with your new hard drive into the PlayStation 4. Slide it in all the way and screw the engraved PlayStation screw back in. Replace the cover.

12. Reconnect the PlayStation 4 to your TV and plug the power chord back in.

13. Your PS4 has two USB ports on the front. Plug the USB Flash Drive into either port and the Dualshock 4 controller into the other one using your controller charge cable (or any other USB cable).

14. Hold the power button (the top of the two hidden buttons located in the "crack" between the two PS4 halves) down for 7 seconds to initiate safe mode.
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15. The Safe Mode options screen will appear. Use your controller to pick the bottom option: Initialize PS4 (Reinstall System Software).
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16. It may take a few minutes for the action to be executed and for the reinstall process to commence. If you get an error message at this point, it is most likely related to the version of the System Install Software you've got on the memory stick (get the latest version of the full install, not the upgrade), or the directory (make sure it's PS4, UPDATE), or the format (FAT32).

Once the install completes, sign back into your PSN account and complete the initial setup. This includes time and date settings, internet/wifi setup, and some basic options. From here, you can restore your game installs from discs and your game saves from a USB stick or from the cloud.

Restoring game saves is done via the Setttings function screen. Select Application Saved Data Management, then select the source (USB or cloud) to restore each title's data. To reinstall games you downloaded from the PlayStation Store, simply head into the store application and access the Library option. You can pick and choose what games to redownload. Note that you sometimes have to dive into the "My DLC" section of each game to download DLC content you've previously installed.
Then you are done!