Circular Domain, Complex Range!

Geometry Level 5

\( \Gamma _ 1 \) is a given circle, with points \( A, B, C \) on the circumference. Let \( \Gamma_2 \) be the circle that passes through the midpoints of sides of triangle \( ABC \). Let \( \ell \) be the length of the common chord of these two circles.

For the combination which maximizes the value of \( \ell \), what would be the corresponding value of

\[ \cos 2A + \cos 2B + \cos 2C \ ? \]

Give your answer correct upto 3 decimal places.

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