Quadratic Equation And Inequalities

Let \(\alpha, \beta\) be the distinct roots of the equation \(x^2-\left(t^2-5 t+6\right) x+1=0, t \in \mathbb{R}\) and \(a_n=\alpha^n+\beta^n\). Then the minimum value of \(\frac{a_{2023}+a_{2025}}{a_{2024}}\) is