\(\displaystyle 1 + \sum_{r = 1}^{10} \left [ 3^r \binom{10}{r} + r \binom{10}{r} \right ] = 2^{10} \left (\alpha \cdot 4^5 + \beta \right ) \)

Consider the above summation, where \(\alpha, \beta \in \mathbb N\) and \(f(x) = x^2 - 2x -k^2 + 1\).

If \(\alpha, \beta \) lies *strictly* in between the roots of \(f(x) = 0\), then find the smallest positive integral value of \(k\).

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