How many step I need to (99,99,19)?

Question

The start point is (0,0,0). The "step vector" is (2,3,4). I can variate this, so it can be (3,4,2) and so on, and it can be negative too, like (4,2-,3). What is the minimum step to reach the finish (99,99,19) point, if I can? And with other step vector, (2,3,5) and (2,3,6)? I tried write a solver program in python but it doesn't work yet.

Answer 1

A useful observation is $$\tag1(-4,3,2)+(3,2,-4)+(2,-4,3)=(1,1,1) $$ and another is $$\tag2(4,3,2)+(3,4,-2)=(7,7,0). $$ Repeating $(2)$ fourteen times, we reach $(98,98,0)$ in $28$ steps, and it is impossible to reach $x+y\ge 196$ in less than $28$ steps. By turning $-2$ into $+2$ five times, we obtain $(98,98,20)$, still in $28$ steps. Whet we still need is $(1,1,-1)$, but that can be done in three steps by flipping all $z$-signs in $(1)$.

Thus we see that it is possible in $31$ steps. Because $99+99+19$ is odd, a solution with an even number of steps is not possible. Hence any better solution would have to consist of $29$ steps (we have already seen that $27$ or less is not possible). Is this possible? I doubt it, but cannot find a smart ans simple argument for that.