Cyclic Equations

For an integer $n > 2$, we have real numbers $a_1, a_2, \cdots, a_n$ such that (here with $n > 4$ for clarity):

$$\begin{align*} a_2 + a_3 + a_4 +\cdots + a_n &= a_1 \\ a_1 + a_3 + a_4 + \cdots + a_n &= a_2 \\ &\vdots \\ a_1 + a_2 + a_3 + \cdots + a_{n-1} &= a_n \end{align*}$$

(In other words, $a_i = \Sigma_{\{j \mid 1 \le j \le n, i \ne j\}} a_j$)

What is the value of $a_1 + a_2 + a_3 + \cdots + a_n$?

(Source: Brilliant (problem page))

This one is as straightforward as adding all the equations together, and seeing what happens. The result will be the equation $(n - 1)(a_1 + a_2 + a_3 + \cdots + a_n) = a_1 + a_2 + a_3 + \cdots + a_n$. (Try it at $n=3$ and $n=4$ for simple concrete examples, if needed.) Whenever, as stated, $n > 2$, the sum of the variables can only ever be zero.