BALKAN-SL 2017 A4

Let $M=\left\{(a, b, c) \in \mathbb{R}^{3}: 0<a, b, c<\frac{1}{2}\right.$ with $\left.a+b+c=1\right\}$ and $f: M \rightarrow \mathbb{R}$ given as

$$f(a, b, c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{a b c}$$

Find the best (real) bounds $\alpha$ and $\beta$ such that

$$f(M)=\{f(a, b, c):(a, b, c) \in M\} \subseteq[\alpha, \beta]$$

and determine whether any of them is achievable.