HMMT 2019 C10

Fred the Four-Dimensional Fluffy Sheep is walking in 4-dimensional space. He starts at the origin. Each minute, he walks from his current position $\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ to some position $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ with integer coordinates satisfying $\left(x_{1}-a_{1}\right)^{2}+\left(x_{2}-a_{2}\right)^{2}+\left(x_{3}-a_{3}\right)^{2}+\left(x_{4}-a_{4}\right)^{2}=4$ and $\left|\left(x_{1}+x_{2}+x_{3}+x_{4}\right)-\left(a_{1}+a_{2}+a_{3}+a_{4}\right)\right|=2$. In how many ways can Fred reach $(10,10,10,10)$ after exactly 40 minutes, if he is allowed to pass through this point during his walk?