Let $f : \mathbb{Z}_{\ge0} \rightarrow \mathbb{Z}_{\ge0}$ be a function satisfying the functional equations: $\ f(3x) = f(x)$, and $f(9x+3a+b)=(a+3b) \cdot f(3x+1)-(a+3b-1) \cdot f(x)\ $ $\forall$ $a \in$ {$0,\ 1,\ 2$}, $b \in$ {$1,\ 2$}, $a+x > 0$. Let $f(0) = 0,\ f(1) = 1,\ f(2) = 2$ and $f(4) = 4$.

Let $g(x)=f(x)-1\ \forall\ x>0$. Find the number of values between 1 and $3^{30}$ for which $f(g^{6}(x))=40$. (Here, $g^{2}(x)=g(g(x))\ne(g(x))^{2}$).