Automatic complexity of Fibonacci and Tribonacci words

For a complexity function $C$, the lower and upper $C$-complexity rates of an infinite word $\mathbf{x}$ are $$\underset{‾}{C}(\mathbf{x})=\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{C(\mathbf{x}\upharpoonright n)}{n},\bar{C}(\mathbf{x})=\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{C(\mathbf{x}\upharpoonright n)}{n}$$ respectively. Here $\mathbf{x}\upharpoonright n$ is the prefix of $x$ of length $n$. We consider the case $C={\mathrm{A}}_{\mathrm{N}}$, the nondeterministic automatic complexity. If these rates are strictly between 0 and $1\mathrm{/}2$, we call them intermediate. Our main result is that words having intermediate ${\mathrm{A}}_{\mathrm{N}}$-rates exist, viz. the infinite Fibonacci and Tribonacci words.