HMMT 2010 U27

Suppose that there are real numbers $a, b, c \geq 1$ and that there are positive reals $x, y, z$ such that

$$\begin{aligned} a^{x}+b^{y}+c^{z} & =4 \\ x a^{x}+y b^{y}+z c^{z} & =6 \\ x^{2} a^{x}+y^{2} b^{y}+z^{2} c^{z} & =9 \end{aligned}$$

What is the maximum possible value of $c$ ?