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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