Prove that $x^4 + 131 = 3y^4$ has no integer solution

Question

So I came across this problem in a book. I figured a few steps out but I am not sure if it is correct.

x⁴ + 131 = 3y⁴
x⁴ + 2 = 3y⁴ - 129
x⁴ + 2 = 3(y⁴ - 43)
x⁴ + 2 ≡ 0 (mod 3)
x⁴ ≡ 1 (mod 3)

Let x⁴ = 3k + 1,
y⁴ = k + 44

I have no idea what to do next, or how to find a contradiction.