Let \(a_{1},a_{2},a_{3},\ldots\ldots,a_{11}\) be real numbers satisfying \(a_{1} = 15,27 - 2a_{2} > 0\) and \(a_{k} = 2a_{k - 1} - a_{k - 2}\) for \(k = 3,4,\ldots,11\). If \(\frac{a_{1}^{2} + a_{2}^{2} + \ldots + a_{11}^{2}}{11} = 90\), then the value of \(\frac{a_{1} + a_{2} + \ldots + a_{11}}{11}\) is equal