A universe consists of an infinite collection of galaxies $G_i$ indexed by the integers. Each galaxy $G_i$ consists of a finite number $g_i$ of stars. Every star in galaxy $G_i$ is connected with every star in galaxy $G_{i-1}$, with every star in galaxy $G_{i+1}$, possibly with some stars in galaxy $G_i$, and with no other stars. It is known that for every sufficiently large $n$ the number $g_{-n}+g_{-n+1}+\ldots +g_{n-1}+g_{n}$ does not exceed $n^{3/2}$. Each star has mass equal to the arithmetic mean (i.e. the average) of the masses of all the stars connected with it. Prove that all stars have the same mass.