IMAS 2022–2023 Junior Division — First Round Question 8

In square $ABCD$, a line through $B$ intersects the extension of $CD$ at $E$, the side $AD$ at $F$ and the diagonal $AC$ at $G$, as shown in the diagram below.



If $BG=3$ cm and $GF=1$ cm, then what is the length, in cm, of $FE$?