Find \(a_{}^{}x^5 + b_{}y^5\) if the real numbers \(a_{}^{}, b_{}^{}, x_{}^{}\), and \(y_{}^{}\) satisfy the equations $$ax + by = 3^{}_{},$$ $$ax^2 + by^2 = 7^{}_{},$$ $$ax^3 + by^3 = 16^{}_{},$$ $$ax^4 + by^4 = 42^{}_{}.$$

(第八届AIME1990 第15题)