How to force Mathematica to simplify this expression?

I have the following expression
$$2^{1-2 n^2} left(3-sqrt{5}right)^{-n-1} left(sqrt{5}-1right)^{n (n+1)} left(sqrt{5}+1right)^{n^2}$$

This can be simplified to
$$left(frac{1}{2} left(sqrt{5}+1right)right)^{n+2}$$

But

 FullSimplify[2^(1 - 2*n^2)*(3 - Sqrt[5])^(-1 - n)*(-1 + Sqrt[5])^(n*(1 + n))*(1 + Sqrt[5])^n^2, n > 0]

returns unevaluated

Interestingly, the ratio of these two expressions is simplified:

FullSimplify[2^(1 - 2*n^2)*(3 - Sqrt[5])^(-1 - n)*(-1 + Sqrt[5])^(n*(1 + n))*((1 + Sqrt[5])^n^2 / ((1 + Sqrt[5])/2)^(n + 2)), n > 0]
(* 1 *)

Simplify[FullSimplify[ExpToTrig[
2^(1 - 2*n^2)*(3 - Sqrt[5])^(-1 - n)*(-1 + 
Sqrt[5])^(n*(1 + n))*(1 + Sqrt[5])^n^2]] /. {1 + Sqrt[5] -> 
2 GoldenRatio, 3 + Sqrt[5] -> 2*GoldenRatio^2}]

 (* GoldenRatio^(2 + n) *)

expr1 = 2^(1 - 2*n^2)*(3 - Sqrt[5])^(-1 - n)*(-1 + 
      Sqrt[5])^(n*(1 + n))*(1 + Sqrt[5])^n^2;
expr2 = expr1 /. (-1 + Sqrt[5])^(n (1 + n)) -> a*(-1 + Sqrt[5])^n^2 //
    Simplify;
expr3 = expr2 /. a -> (-1 + Sqrt[5])^n;
expr4 = expr3 /. (3 - Sqrt[5])^(-1 - n) -> a*(3 - Sqrt[5])^-n // 
   Simplify[#, n > 0] &;
expr5 = expr4 /. a -> (3 - Sqrt[5])^-1;
expr6 = Simplify[expr5/(3 + Sqrt[5]), n > 0]*(3 + Sqrt[5])

 (*   2^(-1 - n) (1 + Sqrt[5])^n (3 + Sqrt[5])   *)