A Golden Fraction II

Number Theory Level 4

$0.1123101\ldots$

Consider a decimal such that the $n^{\text{th}}$ digit to the right of the decimal place is the $n^{\text{th}}$ term of the Fibonacci Sequence $\mod 4$ , as shown above.

The exact value of this decimal can be expressed in the form $\dfrac{a}{b}$ for coprime positive integers $a$ and $b$ .

What is the value of $a+b$ ?