$[4+3$ points$]$ Let $r(t)=c^{t}sinti+c^{t}costj+\sqrt[\phantom{2}]{2}{c}^{t}k$ be a curve.

$($i$)$ Find the arc$-$length function for the curve measured from $P(0,1,\sqrt[\phantom{2}]{2})$ in the direction of increasing $t$ and then reparametrize the curve with respect to arc$-$length starting from the point $P.$

$($ii$)$ Use $($i$)$ to find the position vector of the point $5$ units along the curve from $P.$