Minimum again?

Algebra Level 5

\[\large\left(a+\dfrac{1}{a}\right)^9 +\left(b+\dfrac{1}{b}\right)^9+\left(c+\dfrac{1}{c}\right)^9\]

If \(a,b\) and \(c\) are positive real numbers such that their sum is 1, find the minimum value of the expression above.

If the minimum value can be expressed as \(\dfrac{ w^x}{y^z}\) for positive integers \(w,x,y\) and \(z\), submit your answer as \( w+x+y+z\).

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