X says "according to Y, I always lie." Consider four cases:
Case 1: X = F and Y = F X lies meaning above statement is false and true statement is according to Y, X tells truth. And we have assumed Y lies so according to statement X lies is true (True).
Case 2: X = F and Y = T X lies meaning above statement is false and true statement is according to Y, X tells truth. And we have assumed Y speaks truth so according to statement X speaks truth (contradiction to our assumption---X lies).
Case 3: X = T and Y = F X speaks truth meaning above statement is true so according to Y, X lies. And we have assumed Y lies so X speaks truth is true (True).
Case 4: X = T and Y = T X speaks truth meaning above statement is true so according to Y, X tells lies. And we have assumed Y speaks truth so according to statement X lies (contradiction to our assumption---X speaks truth ).
Answer should be (i and iv)