Is It Really Balancing The Coefficients?

Algebra Level 5

Let \(a\), \(b\) and \(c\) be positive real numbers satisfying \[(a+b+c)\left(\frac {1}{a} + \frac {1}{b} +\frac {1}{c}\right) =16\] If the smallest and largest possible values of \[\frac {a}{b} + \frac {b}{c} +\frac {c}{a} \] are \(m\) and \(M\) respectively, then find the value of \(\lfloor 100 (m+2M )\rfloor\).

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