Two planes $P_1$ and $P_2$ are defined by the equations $x+y+z=1$ and $2x-y+z=0,$ respectively. The planes intersect along a line $L.$

$($a$)$ Write down two vectors which are normal to $P_1$ and $P_2,$ respectively.

$($b$)$ Hence find a vector running in the direction of the line $L.$

$($c$)$ Find the point lying in both planes $P_1$ and $P_2$ which has $z=0.$

$($d$)$ Hence obtain the equation of the line $L.$