Ambiguity by Fibonacci

In the Fibonacci numeral system, integers are represented using Fibonacci numbers as the basis. This sequence starts with $1$, $2$, $3$, $5$, $8$, $13$, $21$, $34$, $55$, $89$, and so on, beginning from $F(2)$. Each integer is expressed as a binary combination, where the least significant (rightmost) digit corresponds to $1$, the next to $2$, and so forth. A digit of $1$ indicates that the corresponding Fibonacci number contributes to the integer's value, while a $0$ means it does not. For instance, the representation (101001)_f translates to the number 1 \cdot 1 + 0 \cdot 2 + 0 \cdot 3 + 1 \cdot 5 + 0 \cdot 8 + 1 \cdot 13 = 19. Unlike the standard binary system, the Fibonacci system allows multiple representations for the same number. For example, the number $19$ can also be represented as (11111)_f.

Given an integer $N$, your task is to determine how many distinct representations it has in the Fibonacci system.