\[ \large \displaystyle \sum_{r=0}^n \frac{r + 1}{{n \choose r}} = \displaystyle \sum_{r=0}^n \frac{n^2 -3n +5 }{2{n \choose r}} \]

If \( n_{1} \) and \(n_{2}\) are the two values of \(n\) which satisfy the given equation above, find the value of \( n_{1}^2 + n_{2}^2\).

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