Complex Numbers and Quadratic Equations: JEE Main 2025 Mathematics question

Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}\,x - 16 = 0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3x - 1 = 0$. If $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$, then $\frac{P_{25} + \sqrt{3}\,P_{24}}{2P_{23}} \;+\; \frac{Q_{25} - Q_{23}}{Q_{24}}$ is equal to: