Example 17 (Problem from the 8th American Invitational Mathematics Examination) If real numbers $a, b, x, y$ satisfy $a x+b y=3, a x^{2}+b y^{2}=$ $7, a x^{3}+b y^{3}=16, a x^{4}+b y^{4}=42$, find the value of $a x^{5}+b y^{5}$.