Question: How to solve a problem in dsolve: unable to convert to an explicit first-order system ?

Hello! I've been trying to solve system of non-linear differential equations. The equations are quite messy, so I've attached the file itself dsolve.mw. I have read about the similar problems, but they all are related to some expressions having multiple branches. I'm not quite sure where this problem occurs (if it does) here and how to solve it.

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