There is an ellipse with focal points

\[P = (4, 2)\]

and

\[Q = (2, 4)\]

One of the points on the ellipse is \((2, 2)\). If the equation for this ellipse is

\[ax^2 + bxy + cy^2 + dx + ey + f = 0\]

where \(\gcd(a, b, c, d, e, f) = 1\) find the maximum value of the following expression:

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

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