Prime Fibonacci? Sounds Crazy!

Fibonacci sequence is defined as $F_0=0 , F_1=1$ and for $n \geq 2$ $F_n=F_{n-1}+F_{n-2}$

So the Fibonacci sequence is $0,1,1,2,3,5,8,13,...$

Find the sum of all the terms in the Fibonacci sequence which are less than $\textbf{1 billion}$ and are $\textbf{prime numbers}$.

Details and assumptions:-

$\bullet$ A prime number is the number which has only $2$ positive integer divisors, which are $1$ and the number itself. No other positive integer divides the number.

$\bullet$ 1 is not a prime number, 2 is the only even prime number.

This problem is a part of the set Crazy Fibonacci