If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $. I solved it using AM, GM inequalities and reached to $[(1+a)(1+b)(1+c)]^7 > 2^{21}(abc)^\frac72 $ please help how to get $7^7(abc)^4$ in the inequality.

Here we have to prove $$(1+a)(1+b)(1+c)>7(abc)^{\frac{4}{7}}$$
So we get $$1+(a+b+c)+(ab+bc+ca)+abc>7(abc)^{\frac{4}{7}}$$
Now Using $\bf{A.M\geq G.M}\;,$ We get $$a+b+c\geq 3(abc)^{\frac{1}{3}}$$
and $$ab+bc+ca\geq 3(abc)^{\frac{2}{3}}$$
So $$1+\sum a+\sum ab+abc\geq 1+3(abc)^{\frac{1}{3}}+3(abc)^{\frac{2}{3}}+abc$$
Now Again Using $\bf{A.M\geq G.M}\;,$ We get
$$\bf{L.H.S}>1+(abc)^{\frac{1}{3}}+(abc)^{\frac{1}{3}}+(abc)^{\frac{1}{3}}+(abc)^{\frac{2}{3}}+(abc)^{\frac{2}{3}}+(abc)^{\frac{2}{3}}+(abc)\geq 7(abc)^{\frac{4}{7}}$$
