A relation between the distance \(r\) of any point on a given curve from the origin and the length of the perpendicular \(p\) from the origin to the tangent at that point is called **Pedal Equation** of the curve.

Consider a curve represented by the equation:

\[
c^{2}(x^{2} + y^{2}) = x^{2}y^{2}
\]

where \(c\) is some constant.

If the Pedal Equation of this curve can be written in the form:

\[
\frac{\alpha}{p^{\beta}} + \frac{\gamma}{r^{\beta}} = \frac{\alpha}{c^{\beta}}
\]

where \(\alpha\) , \(\beta\) , \(\gamma\) are positive coprime integers. Find the value of \(\alpha+\beta+\gamma\)

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