$(15$p$.)$ Assume $x$ and $Y$ are independent exponential random variables with rate parameter $1.$ For the function

$g(x,y)=e^{-x-y},$

a$.$ Find $E[g(x,Y)|Y]$

b$.$ Find $E[g(x,Y)|x]$

c$.$ Find $E[g(x,Y)]$