IMC 2024 Problem 8

Define the sequence $x_1,x_2,\dots$ by the initial terms $x_1=2, x_2=4$ , and the recurrence relation $$x_{n+2}=3x_{n+1}-2x_n+\frac{2^n}{x_n} \text{for} n \ge 1.$$ Prove that $\lim_{n \to \infty} \frac{x_n}{2^n}$ exists and satisfies $$\frac{1+\sqrt{3}}{2} \le \lim_{n \to \infty} \frac{x_n}{2^n} \le \frac{3}{2}.$$