Find the sum of all the Fibonacci numbers $F_n$ less than 1 billion which follow the condition $13 \mid (4 F_n^2 -22F_n+11)$

**Details and assumptions**:

Fibonacci sequence is defined as $F_0=0,F_1=1$ and for $n\geq 2$, $F_n=F_{n-1}+F_{n-2}$. Thus, the Fibonacci sequence is $0,1,1,2,3,5,8,13,\ldots$.

$F_n$ denote the $n$-th Fibonacci number.