Let \( \alpha, \beta \) be the roots of the equation \( x^2 - ax - b = 0 \) with \( \operatorname{Im}(\alpha) < \operatorname{Im}(\beta) \). Let \( P_n = \alpha^n - \beta^n \). If \( P_3 = -5\sqrt{7}\,i \), \( P_4 = -3\sqrt{7}\,i \), \( P_5 = 11\sqrt{7}\,i \) and \( P_6 = 45\sqrt{7}\,i \), then \( |\alpha^4 + \beta^4| \) is equal to __________.
