You are given a directed graph $G$ with edges labeled $0,1$ or $2.$ You have a starting vertex $s$ and an ending vertex $t.$

You wish to find the length of the shortest walk from $s$ to $t($fewest number of edges$)$ such that in order to traverse an edge labeled $2,$ you must first traverse at least one edge labeled $1.$

For example, the shortest walk from $s$ to $t$ with the above conditions is $6$ from the walk: $(s,A,B,C,A,B,t).$

$(8$ points for algorithm description. High$-$level is required, implementation$-$level or pseudocode is optional.$)$

$(3$ points for efficiency $O(|V|+|E|))$

$(5$ points runtime justification.$)$