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 $(a_1,\ a_2,\ a_3,\ a_4)$ to some position $(x_1,\ x_2,\ x_3,\ x_4)$ with integer coordinates satisfying $$(x_1−a_1)^2+(x_2−a_2)^2+(x_3−a_3)^2+(x_4−a_4)^2 = 4$$

and $\mid (x_1 + x_2 + x_3 + x_4) − (a_1 + a_2 + a_3 + a_4)\mid = 2$. 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?