Prove that if $ a,b,c > 0 $, then $ [(1 + a) (1 + b) (1 + c)]^{7} > 7^{7} (a^{4} b^{4} c^{4}) $. 
Problem. Prove that if $ a,b,c > 0 $, then $ [(1 + a) (1 + b) (1 + c)]^{7} > 7^{7} (a^{4} b^{4} c^{4}) $.

I don’t know how to solve this problem... What I can think of is to just simplify this inequality:
$$
\left[ \frac{(1 + a) (1 + b) (1 + c)}{7} \right]^{7} > a^{4} b^{4} c^{4}.
$$
How can I proceed with solving this problem?  
Note: This is a question of sequence and series, specifically AM-GM-HM inequality...

Notice 
$$\frac{(1+a)(1+b)(1+c)-1}{7}=\frac{a+b+c+ab+bc+ca+abc}{7} \ge \sqrt[7]{a^4b^4c^4}$$
Because of $\text{AM-GM}$. 
We conclude 
$$\frac{(1+a)(1+b)(1+c)}{7}>\frac{(1+a)(1+b)(1+c)-1}{7} \ge\sqrt[7]{a^4b^4c^4}$$
This is equivalent to 
$$
\left[ \frac{(1 + a) (1 + b) (1 + c)}{7} \right]^{7} > a^{4} b^{4} c^{4}.
$$
