Let $k$ be the largest integer such that $2^k$ divides $$\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}\right)^2\right)\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}^2\right)\right).$$ Find $k$.

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