Consider the system of equations given by:
\[ x + y + z = 6 \]
\[ xy + yz + zx = 11 \]
\[ xyz = 6 \]
Additionally, in a triangle, the angles are given by \( A = x^\circ \), \( B = y^\circ \), and \( C = z^\circ \). Verify if such a triangle can exist and, if so, find the value of \( \cos A + \cos B + \cos C \). Let's think step by step and output the final answer within \boxed{}.
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