For any complex number \(\omega = a+bi\), \( |\omega| \) is defined to be the real number \(\sqrt{a^2+b^2}\). If \(\omega =\cos 40^\circ + \cos 40^\circ\), then find\( |\omega +2\omega^2 +3\omega^3 + ... + 9\omega^9|^{-1}.\) Express your answer to the first three decimal places.

This problem is not original.

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