Does the laptop have a third party NVMe SSD installed? If so, then you need to boot a macOS 10.13+ installer since older versions of macOS don't have the necessary NVMe driver required for the third party SSD. Try booting into Internet Recovery Mode using Command + Option + R to attempt to boot into the online macOS 10.15 Catalina or 11.x Big Sur installer depending on whether you have the early or late 2013 model respectively.
Even better, I highly recommend creating a bootable macOS USB installer using the instructions in this Apple article while your laptop is still able to boot normally just in case something goes wrong using Internet Recovery Mode.
How to create a bootable installer for macOS - Apple Support
Also before performing the clean install, review this Apple article on other things you may need to do before erasing the Mac:
What to do before you sell, give away, or trade in your Mac - Apple Support