True or false or it depends?

Algebra Level 3

True or false:

If \(a\left(\dfrac{1}{b} + \dfrac{1}{c}\right) , b\left(\dfrac{1}{c} + \dfrac{1}{a}\right) , c\left(\dfrac{1}{a} + \dfrac{1}{b}\right)\) are in arithmetic progression where \(a,b,c\) are real numbers not equal to 0 then \(a^3 + c^3 + 6abc\) equal to \(8b^3\).

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