Shell: how to split a file with blocks as delimiters

Hi,

I have a file.txt:
Quote:

a b c d e f g h i

I would like to split the file when I met a fixed number or more of consecutive blank lines.
So with 3+ blank lines as a separator, I'd get:

file1.txt:
Quote:

a b c d e f

file2.txt:Quote:

g

file3.txt:Quote:

h i

I have various ideas with counters but I have an efficiency constraint there, and I lack knowledge about advanced features of sed/awk-likes.
So any help is welcome.