The cosine summation

Geometry Level 4

The value of r=012(12r)cosrπ6\displaystyle\sum_{r=0}^{12} \binom{12}{r} \cos \frac {r\pi}{6}

is of the form(x+y)z-(x+\sqrt {y})^{z}, where x,yx,y and zz are integers, and zz is as large as possible.

Find x+y+zx+y+z.

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