\[ \large \dfrac14 \int_0^1 \dfrac{t (1-t)}{1+t^2(1-t)^2} \, dt = \Re(-z \tan^{-1} z ) \]

If the equation above holds true, where \(z\) is a zero of a polynomial with integer coefficients, find \( |z|^{-4} \). \[\]

**Clarification**: \( \Re(x) \) denotes the real part of the complex number \(x\). For example, \(\Re(3+4i) = 3 \).

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