For $n=1,2, \ldots$ let
$$
S_{n}=\log \left(\sqrt[n^{2}]{1^{1} \cdot 2^{2} \cdot \ldots \cdot n^{n}}\right)-\log (\sqrt{n})
$$
where $\log$ denotes the natural logarithm. Find $\lim\limits _{n \rightarrow \infty} S_{n}$. The original answer is in $-\frac{m}{n}$ format, please give the value of m + n.

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