KöMaL 2025 B5487

Positive numbers $a_1$, $a_2$, $\dots$, $a_{2025}$ satisfy $a_1=1$ and $\frac{1}{a_1+a_2}+\frac{1}{a_2+a_3}+\ldots+\frac{1}{a_{n-1}+a_n}=a_n-1$ for every $2\le n\le 2025$. Find the value of $a_{2025}$.