Consider the following sequence \(\left(a_{n}\right)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1, \ldots)\). Find all pairs \((\alpha, \beta)\) of positive real numbers such that \(\lim \limits_{n \rightarrow \infty} \frac{\sum\limits_{k=1}^{n} a_{k}}{n^{\alpha}}=\beta\). The original answer is in \(\frac{\sqrt{m}}{n}\) 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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