Grab first or second line after a search string

In a shell script, I need to grab the first or second line after a search string in a file. For example:

File.out:

Random Info
Manufacturer: XYZPDQ

System Info
Manufacturer: Hewlett-Packard
Product Name: ProLiant

I search for the word FILE, I want to be able to grab the line after that, without having to search for a string in that first line.

if you have GNU grep, check the -A and -B (and also -C) option