$(2)$ Study the system

$x+2my+z=4m$

$2mx+y+z=2$

$x+y+2mz=2m^2$

with real pormeter $m.$ You must determine, with proof, for which $m$ this system has

$($i$)$ no solution,

$($ii$)$ exactly one solution,

$($iii$)$ infinitely many solutions.