Two applications of the implicit function theorem to a nonlinear system

Consider \[ 3x+y-z+u^4=0, \qquad x-y+2z+u=0, \qquad 2x+2y-3z+2u=0. \]

1. Prove that for some \(\varepsilon>0\) the system can be solved for \((x,y,u)\) as functions of \(z\in[-\varepsilon,\varepsilon]\), with \(x(0)=y(0)=u(0)=0\). Are these functions continuous, differentiable, and unique?
2. Show that the system cannot be solved for \((x,y,z)\) as functions of \(u\in[-\delta,\delta]\) for every \(\delta>0\).