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