3. Given the line $y=-\sqrt{3} x+2 \sqrt{3}$ intersects the $x$-axis and $y$-axis at points $B$ and $C$ respectively, point $A(-2,0)$, and $P$ is a moving point on line $B C$. Translate $B C$, when $B C$ is in different positions, the number of points $P$ that satisfy $\angle A P O=30^{\circ}$ is $\qquad$ cases.
3. 5 .

By the inscribed angle theorem, we know that point \( P \) lies on the arcs \(\overparen{A M O}\) and \(\overparen{A N O}\), as shown in Figure 4.

These two major arcs are parts of the upper and lower equal circles with \( A O \) as the chord and the central angle subtended by \( A O \) equal to \( 60^{\circ} \). By translating \( B C \), we know that the number of points \( P \) can be \( 0, 1, \cdots, 4 \), for a total of 5 cases.