$ 3^7\cdot (a^9+b^9+c^9)+1\geq 12\cdot (a^3+b^3+c^3)$ Let $a,b,c>0$ s.t. $a+b+c=1$. Show that $$ 3^7\cdot (a^9+b^9+c^9)+1\geq 12\cdot (a^3+b^3+c^3)$$
I tried to apply the next formula:$$a^3+b^3+ c^3=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)+3abc$$
Also I made substitutions $ab+bc+ac=x$ and $abc=y$ but I am stuck.

Hint: Show that $ 3^7 a^9 + a \geq 12 a^3$ using AM-GM.
Corollary: The result follows by summing up the cyclic inequalities.
