Consider two unit vectors \( \vec{u}_1, \vec{u}_2 \) chosen uniformly at random from all unit vectors in \( \mathbb{R}^3 \) such that \( \vec{u}_1 \cdot \hat{k} = \vec{u}_2 \cdot \hat{k} = 0 \). What is the expected value of \( |u_1 + u_2| \)?

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