# 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 $$a+b+c+d$$.

Inspiration

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