\[\large\color{darkblue}{\mathfrak{N}}=\displaystyle\sum_{m=0}^{2^{2^{5}}-1} \left( \dfrac{1+1}{\displaystyle\prod_{n=1}^{5}\left(\sqrt[2^n]{m+2}+\sqrt[2^n]{m}\right)}\right)\]

\[\large\log_2\left(\log_{2}\left((\color{darkblue}{\mathfrak{N}}-1)^{2^5}-1\right)\right)+4=\, ?\]

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