Hard problem

Algebra Level 4

If $$a+b=1$$, the minimum value of

$\left(a+\frac{1}{a}\right)^2+\left(b+\frac{1}{b}\right)^2$

can be expressed in the form $$\frac{m}{n}$$, where $$m,n$$ are coprime, positive integers. Find the value of $$m+n$$.

$$\textbf{(This problem is not original.)}$$

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