iPhone could not be restored failed for unknown error 9. How to reset?

My iPhone 5s started rebooting randomly, then started a reboot cycle that never went into the home location. When I plugged it into the Mac, with iTunes running, it still kept rebooting and the iTunes did not see it. I had to hold down the Home button during one of the reboots for iTunes to see it.

It stated there was an issue and the phone needed to be restored and updated. I followed the process, which led to:

"The iPhone "iPhone" could not be restored. An unknown error occurred (9).

I tried the restore-and-update after each of these actions:

• Updating iTunes
• Restarting both Mac and iPhone
• Following the instructions on support.apple.com/iphone/restore
• Hard boot of the iPhone both while unconnected and connected

It seems like the phone needs a basic restore before doing an update, and not both at the same time. How can I do a restore to get the phone back to a good state? Once there, I am sure the update would work.