Let \( \alpha, \beta \; (\alpha \ne \beta) \) be the values of \( m \), for which the equations \( x + y + z = 1 \); \( x + 2y + 4z = m \) and \( x + 4y + 10z = m^{2} \) have infinitely many solutions. Then the value of \( \sum\limits_{n=1}^{10} \left(n^{\alpha} + n^{\beta}\right) \) is equal to:
