1175. Death to Binary?

The group of Absurd Calculation Maniacs has discovered a great new way how to count. Instead of using the ordinary decadic numbers, they use Fibonacci base numbers. Numbers in this base are expressed as sequences of zeros and ones similarly to the binary numbers, but the weights of bits (fits?) in the representation are not powers of two, but the elements of the Fibonacci progression ($1,2,3,5,8,\dots$ - the progression is defined by $F_0=1,F_1=2$ and the recursive relation $F_n=F_{n-1}+F_{n-2}$ for $n\ge 2$).

For example ${1101001}_{Fib}=F_0+F_3+F_5+F_6=1+5+13+21=40$.

You may observe that every integer can be expressed in this base, but not necessarily in a unique way - for example $40$ can be also expressed as ${10001001}_{Fib}$. However, for any integer there is a unique representation that does not contain two adjacent digits $1$ - we call this representation canonical. For example ${10001001}_{Fib}$ is a canonical Fibonacci representation of 40.

To prove that this representation of numbers is superior to the others, ACM have decided to create a computer that will compute in Fibonacci base. Your task is to create a program that takes two numbers in Fibonacci base (not necessarily in the canonical representation) and adds them together.