How many strings of length $4$ formed from the digits $0,1,2,3$ are there such that for each $j\in 1,2,3$, at least $j$ of the digits are less than $j$? (For example, $0012$ satisfies this condition because it contains at least $1$ digit less than $1$, at least $2$ digits less than $2$, and at least $3$ digits less than $3$. The string $3321$ does not satisfy the condition because it does not contain at least $1$ digit less than $1$.)