\[\Large \mathfrak P=\displaystyle\int_0^{\pi/2}\mathfrak{T}(\phi)\mathrm{d}\phi\]

Given \(\large \mathfrak{T}(\phi)=\displaystyle\sum_{n=0}^{\infty}\dfrac{\cos^3\phi}{\csc^{4n} \phi},~(|\csc \phi|\neq 1)\), if the value of \(\mathfrak P\) can be written in the form \(\large\dfrac{\pi}{\psi^{\psi}}\) \(~~(\psi\in\mathbb{Z}^+)\) , then: \[\Large\color{red}{\psi^{\psi^{\psi+1}}}=\ ?\]

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