Inequality : $\frac{a^{n+1}}{x}+\frac{b^{n+1}}{y}+\frac{c^{n+1}}{z}\geq1 $ 
Prove that if $a, b, c, x, y, z >0$ and $n$ is positive integer such that
$(a^n+b^n+c^n)^{n+1}=x^n+y^n+z^n$ , then $$\frac{a^{n+1}}{x}+\frac{b^{n+1}}{y}+\frac{c^{n+1}}{z}\geq1 $$

My attempt :
By Holder Inequality,
$$(a^{n+1}+b^{n+1}+c^{n+1})^n(1+1+1) \geq (a^n+b^n+c^n)^{n+1}$$
$$(a^{n+1}+b^{n+1}+c^{n+1})^n \geq \frac{x^n+y^n+z^n}{3}$$
I don't know how to proceed.

Hint: Holder: 
$$
(x^n+y^n+z^n)^{1/n+1}(\frac{a^{n+1}}x+\frac{b^{n+1}}y+\frac{c^{n+1}}z)^{n/n+1}>?
$$
