VTRMC 2004 Problem 2

A sequence of integers $\{f(n)\}$ for $n=0,1,2,\ldots$ is defined as follows: $f(0)=0$ and for $n>0$ , $$\begin{matrix}f(n)=&f(n-1)+3,&\text{if }n=0\text{ or }1\pmod6,&f(n-1)+1,&\text{if }n=2\text{ or }5\pmod6,&f(n-1)+2,&\text{if }n=3\text{ or }4\pmod6.\end{matrix}$$ Derive an explicit formula for $f(n)$ when $n\equiv0\pmod6$ , showing all necessary details in your derivation.