Given $n=2020$, sort the 6 values: $$n^{n^2},\, 2^{2^{2^n}},\, n^{2^n},\, 2^{2^{n^2}},\, 2^{n^n},\, \text{and}\, 2^{n^{2^2}}$$ from least to greatest. Provide your answer as a 6-digit permutation of the string "123456", where the number $i$ corresponds to the $i$-th expression in the list, from left to right.

Let's think step by step and output the final answer within \boxed{}.
user