Inequality $\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{a}{ac+a+1}\ge \frac{3m}{{m}^{2}+m+1}$

Let $m=(abc{)}^{\frac{1}{3}}$, where $a,b,c\in {\mathbb{R}}^{\mathbb{+}}$. Then prove that

$\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{a}{ac+a+1}\ge \frac{3m}{{m}^{2}+m+1}$

In this inequality I first applied Titu's lemma ; then Rhs will come 9/(some terms) ; now to maximise the rhs I tried to minimise the denominator by applying AM-GM inequality.But then the reverse inequality is coming Please help.

Let $m=(abc{)}^{\frac{1}{3}}$, where $a,b,c\in {\mathbb{R}}^{\mathbb{+}}$. Then prove that

$\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{a}{ac+a+1}\ge \frac{3m}{{m}^{2}+m+1}$

In this inequality I first applied Titu's lemma ; then Rhs will come 9/(some terms) ; now to maximise the rhs I tried to minimise the denominator by applying AM-GM inequality.But then the reverse inequality is coming Please help.