Consider a $4\times 4$ matrix $A$ with eigenvectors $u,v,w,x$ corresponding to distinct eigenvalues $\lambda{_1},\lambda{_2},\lambda{_3},\lambda{_4}$ where $1 = \lambda{_1} > \lambda{_2} > \lambda{_3} > \lambda{_4} > 0$. Consider a vector $z = 2u + v - w + 3x$. Then, $\lim_{k\rightarrow \infty} A^kz = $_____.
