Looks easy but still Dangerous.

Calculus Level 5

A curve passing through the origin, is such that the middle point of the segment of normal between point of contact and \(x\)-axis lies on the parabola \(2y^2 = x\), then the equation the curve is \(a_{1}y^{a_{2}} = a_{3}x + a_{4} - a_{5}e^{a_{6}x}\). Find the value of \(a_{1} +a_{2} + a_{3} +a_{4} + a_{5} + a_{6}\).

Given that \(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}\) are integers which may or may not be equal with \(a_{1}\) > 0 and their HCF is 1.

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