For the equations

$$\begin{gathered} p x+y+z=1 \\ x+2 y+4 z=t \\ x+4 y+10 z=t^{2} \end{gathered}$$

find the values of $p$ and $t$ for which

(i) there is a unique solution;

(ii) there are infinitely many solutions;

(iii) there is no solution.