Straight Lines Really Are Straight
The line \(4x + y = 1\) passes through the point \(A(2,-7)\) and intersects point \(B\) on the line \(BC\) whose equation is \(3x - 4y + 1=0\). The equation of the line \(AC\) is \(ax + by + c = 0\) and constructed so that the distance from \(A\) to \(B\) is the same as the distance from \(A\) to \(C\).
What is the minimum positive value of \(a+b+c\)?
Note: \(A,B,C\) are all different points in the coordinate plane.