ISL 1993 Problem 9

Let $a,b,c,d$ be four non-negative numbers satisfying $$a+b+c+d=1.$$ Prove the inequality $$a \cdot b \cdot c + b \cdot c \cdot d + c \cdot d \cdot a + d \cdot a \cdot b \leq \frac{1}{27} + \frac{176}{27} \cdot a \cdot b \cdot c \cdot d.$$