As I already explained in the comments, the monitor telling you "(input) out of range" means that your graphics adapter's output is out of the range your monitor can handle. Since that is apparently already the case in the grub environment, you should change grub's graphic configuration to a value that both your graphics adapter and monitor can handle.
The usual place to configure grub is
/etc/default/grub, and you should always use that file if possible. However, you'll need to run
update-grub to compile the new configuration, which can be tedious from the outside (e.g. a live system). So if you are only able to access the file system from the outside, it's easier to directly modify
/boot/grub/grub.cfg. In your case, find the
set gfxmode= line and set it to something like
If that worked and you are able to boot and use the system, do the same change in
/etc/default/grub (the setting is called
GRUB_GFXMODE there and usually commented out). Otherwise, the next
update-grub will overwrite your changes (for example when a new kernel has been installed).