4. For the acute triangle $\qquad$ $ABC$, find
$$
F=\frac{\sin A+\sin B+\sin C}{\cos A+\cos B+\cos C}
$$

the range of values for $F$.
Given $0<\angle A \leqslant \angle B \leqslant \angle C<\frac{\pi}{2}$. Then $0<\angle A \leqslant \frac{\pi}{3} \leqslant \angle C<\frac{\pi}{2}$.
Fixing $\angle C$ we get
$F=\frac{2 \cos \frac{C}{2} \cdot \cos \frac{A-B}{2}+\sin C}{2 \sin \frac{C}{2} \cdot \cos \frac{A-B}{2}+\cos C}$.
Let $f(x)=\frac{2 x \cos \frac{C}{2}+\sin C}{2 x \sin \frac{C}{2}+\cos C}$.
We can find the minimum value of $f(x)$.
Similarly, fixing $\angle A \in\left(0, \frac{\pi}{3}\right]$, we get
$F=\frac{2 \cos \frac{A}{2} \cdot \cos \frac{B-C}{2}+\sin A}{2 \sin \frac{A}{2} \cdot \cos \frac{B-C}{2}+\cos A}$.
Let $g(x)=\frac{2 x \cos \frac{A}{2}+\sin A}{2 x \sin \frac{A}{2}+\cos A}$, we can find the maximum value of $g(x)$. Thus, we can determine the range $\left(1+\frac{\sqrt{2}}{2}, 2\right)$.