Let $a,b,c,d,e,f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of

$\color{white}\ .\quad \ \color{black}\ \quad abc+bcd+cde+def+efa+fab$

and determine all $6$-tuples $(a,b,c,d,e,f)$ for which this maximal value is achieved.

Let's think step by step and output the final answer within \boxed{}.
user