Question 6
Find the largest real number $k$ such that for all positive real numbers $a,b,c$ with $abc=1$, the inequality
$$\frac1{a^2}+\frac1{b^2}+\frac1{c^2}+3k\ge(k+1)(a+b+c)$$
always holds.
$$\frac1{a^2}+\frac1{b^2}+\frac1{c^2}+3k\ge(k+1)(a+b+c)$$
always holds.