Determine all possible pairs (x,y) of positive integers with gcd(4x+1, 4y-1) = 1 such that x+y divides 16xy+1

(In reply to

re: Two unhelpful(?) observations by broll)

Well, I believe I proved my observation, but it is not true that the observation rules out solutions.

For instance, (1,2) is a solution.

The gcd(4x+1,4y-1) = gcd(5,7) = 1

and x+y = 3 divides 16xy+1=33

Also, per my observation,

x+y = 3 divides both 5*3 and 9*7