I Dare You To Take A Special Case! Part 2

Let $\Gamma_1$ be the circumcircle of $\Delta ABC$ and $\Gamma_2$ be the circumcircle of $\Delta DEF$ where $D,E,F$ are the foot of the altitudes of $\Delta ABC$ .

If $\Gamma_1$ cuts $\Gamma_2$ orthogonally, compute the value of the following expression upto three decimal places:

$\sum_\text{cyc} \cos 2A$

Notations:

• $A$ denotes $\angle BAC$ . $B$ and $C$ are defined similarly.
• $\displaystyle \sum_\text{cyc}$ denotes cyclic summation.

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This problem is part of my set: Geometry