Value from a Complex Quotient

Given that $\frac{(1+i)^{2025}}{(1-i)^{2026}} = a + bi$ where $a$ and $b$ are real numbers, find the value of $a^2$. The answer is in the form $\frac{\alpha}{\beta}$ for coprime integers $\alpha$ and $\beta$. Input the answer as the sum $\alpha +\beta$.