$4$ Unprogrammable Programs

Prove whether the programs described below can exist or not.

$($a$)$ A program $P(F,x,y)$ that returns true if the program $F$ outputs $y$ when given $x$ as input $($i$.$e$.F(x)=y$ and false otherwise.

$($b$)$ A program $P$ that takes two programs $F$ and $G$ as arguments, and returns true if $F$ and $G$ halt on the same set of inputs $($or false otherwise$).$

Hint: Use $P$ to solve the halting problem, and consider defining two subroutines to pass in to $P,$ where one of the subroutines always loops.