The line \(x+y = 1 \) in \(xy\)-plane bisects **two distinct** chords of a standard parabola (which is symmetric about the \(x\)-axis and whose vertex is the origin having latus rectum length \(= 4a\)). If it is given that the intersection point of the two chords is \((a,2a)\), find the sum of all integral values of the possible lengths of latus rectum of the parabola.

**Details and Assumptions**

- Latus rectum is the focal chord of the parabola which is perpendicular to its axis.
- If your answers are 5, 6 and 7 then provide the answer as \(5+6+7 = 18\).
- If you think no such parabola is possible give the answer as 0.

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