There are enough points, but how do you determine the equation?

Algebra Level 5

There is a hyberbola that passes through points

\[A = (0, 6)\]

\[B = (1, 3)\]

\[C = (3, 1)\]

\[D = (6, 0)\]

\[E = (10, -1)\]

If the equation for this hyperbola can be represented as

\[ax^2 + by^2 + cxy + dx + ey + f = 0\]

where the \(\gcd (a, b, c, d, e, f)\) is 1, find the minimum value of

\[a + b + c + d + e + f\]

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