Find the value of: cot [Σ_(n=1)²⁵ cot⁻¹ (1 + Σ_(k=1)ⁿ 2k)].

$\cot\left[\sum_{n=1}^{25}\cot^{-1}\left(1 + \sum_{k=1}^{n}2k\right)\right] = \cot\left[\sum_{n=1}^{25}\cot^{-1}\left(1 + 2\times\frac{n(n+1)}{2}\right)\right]$ $= \cot\left[\sum_{n=1}^{25}\cot^{-1}(n^2 + n + 1)\right]$ $= \cot\left[\sum_{n=1}^{25}\tan^{-1}\left(\frac{1}{n^2+n+1}\right)\right]$ $= \cot\left[\sum_{n=1}^{25}\tan^{-1}\left(\frac{n+1-n}{1+n(n+1)}\right)\right]$ $= \cot\left[\sum_{n=1}^{25}\left(\tan^{-1}(n+1) - \tan^{-1}(n)\right)\right]$ $= \cot\left[\tan^{-1}(26) - \tan^{-1}(1)\right]$ $= \cot\left[\tan^{-1}\frac{25}{27}\right] = \cot\left[\cot^{-1}\frac{27}{25}\right] = \frac{27}{25}$