Systems of Differential Equations · Hard problem

Let $A=\begin{pmatrix}2&1&1\\1&2&1\\1&1&2\end{pmatrix}$ and $\mathbf x'=A\mathbf x$ with $\mathbf x(0)=(1,0,0)$. As $t\to\infty$, $x_1(t)$ grows like $Ce^{4t}$ for a constant $C$. Find $C$.