Let $k$ be an integer greater than 1. Suppose $a_0>0$, and define $$a_{n+1}=a_n+\frac{1}{\sqrt[k]{a_n}}$$ for $n>0.$ Evaluate $$\lim_{n\to\infty}\frac{a_n^{k+1}}{n^k}.$$ The original answer is in \left(\frac{k+1}{k}\right)^{k} format, please give the value of m + n where m/n is the simplified fraction form of the original answer when k = 1.

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