4. If the system of inequalities about $x$ $\left\{\begin{array}{l}x^{3}+3 x^{2}-x-3>0, \\ x^{2}-2 a x-1 \leqslant 0\end{array} \quad(a>0)\right.$ has exactly one integer solution, then the range of values for $a$ is $\qquad$.
Solve:
$$
x^{3}+3 x^{2}-x-3>0 \Rightarrow(x+3)(x+1)(x-1)>0 \Rightarrow x \in(-3,-1) \cup(1,+\infty) \text {. }
$$

When $a>0$, the axis of symmetry of the graph of $f(x)=x^{2}-2 a x-1$ is $x=a>0$ and $f(0)=-10, \\ f(2)=4-4 a-1 \leqslant 0, \\ f(3)=9-6 a-1>0\end{array}\right.$ $\Rightarrow \frac{3}{4} \leqslant a<\frac{4}{3}$. Therefore, the range of $a$ is $\left[\frac{3}{4}, \frac{4}{3}\right)$.