Given the finite field GF$(23),$ here are two elements in the field, A$($x$)=$x$2+$x$+1$ and B$($x$)=$x$2.$ Calculate the following equations based on the arithmetic in the finite field GF$(23)$:

$(1)$ A$($x$)+$B$($x$)$; $(5$ points$)$

$(2)$ Given p$($x$)=$x$3+$x$+1,$ what is A$($x$)$B$($x$)$mod$$p$($x$)?(5$ points$)$