IMC 2024 Problem 2

For $n=1,2,\dots$ let $$S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}),$$ where $\log$ denotes the natural logarithm. Find $\lim_{n \to \infty} S_n$ .