My last problem of the year

Geometry Level 5

\[\large{\dfrac{2\sqrt{13}}{3}}\cos\left(\dfrac{\pi}{3}-\dfrac{1}{3}\arctan\left(\dfrac{3\sqrt{3}}{5}\right)\right)-\dfrac{1}{3}\] If the value of the expression above can be expressed as: \[\large{a\left(\cos\left(\dfrac{b \pi}{d}\right)+\cos\left(\dfrac{c \pi}{d}\right)\right)}\] where \(a,b,c,d\) are positive integers, \(\gcd(b,d)=\gcd(c,d)=1\) and all the angles belong to \([0,\pi]\), find \(a+b+c+d\).


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