There is a real number $x$ between $0$ and $\frac{\pi}{2}$ such that $$\frac{\sin^3 x + \cos^3 x}{\sin^5 x + \cos^5 x}=\frac{12}{11}$$ and $\sin x + \cos x =\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers, and $m$ is not divisible by the square of any prime. Find $m + n$.

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