The sequence of positive integers $a_1, a_2, \ldots, a_{2025}$ is defined as follows:
- $a_1 = 2^{2024} + 1$
- For $1 \leq n \leq 2024$, $a_{n+1}$ is the greatest prime factor of $a_n^2 - 1$.

Find the value of $a_{2024} + a_{2025}$.

Let's think step by step and output the final answer within \boxed{}.
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