Example 1 Given any 5 points on a plane, where no 3 points are collinear and no 4 points are concyclic, if a circle passes through 3 of these points and the other two points are one inside and one outside the circle, it is called a "good circle". If the number of good circles is denoted as $n$, find all possible values of $n$.
(1991 China National Training Team Selection Exam Problem)
Consider any two points $A, B$ among the 5 points, and draw a line through $A, B$. If the other three points $C, D, E$ are on the same side of line $AB$, then examine $\angle ACB, \angle ADB, \angle AEB$. Without loss of generality, assume $\angle ACB < 180^{\circ}$. Then the circle $ADB$ is the unique good circle; if $\angle AEB + \angle ADB > 180^{\circ}$, and $\angle AEB + \angle ACB < 180^{\circ}$, then the circle $AEB$ is the unique good circle. 

If $n = 5$, connect every two points among the 5 points with a line segment. Each line segment is either a chord of a good circle or a common chord of 3 good circles. If there are exactly 5 good circles, since each good circle has 3 chords, they have a total of 15 chords. Since there are 10 line segments in total, and each line segment contributes 1 or 3 to the above count, but the sum of 10 odd numbers is even, it cannot equal 15, so $n \neq 5$. 

If there are at least 6 good circles, since each circle has 3 chords, there are a total of 18 chords. In this case, the number of line segments contributing 3 to the count is at least 4. 4 line segments have 8 endpoints, so there must be two line segments with a common endpoint, say $AB, AC$. Then the circles $ABD, ACD$ are both good circles, which means there are at least 2 good circles passing through $A, D$, and of course 3. Similarly, there are 3 good circles passing through $A, E$, so there are 3 good circles passing through $AB, AC, AD, AE$.

Let the shortest side among $AB, AC, AD, AE$ be $AB$. Then $\angle ACB, \angle AEB, \angle ADB$ are all acute angles. If $C, D, E$ are on the same side of $AB$, then there is only 1 good circle passing through $AB$, which is a contradiction. So these 3 points must be on opposite sides of $AB$. In this case, since any two of $\angle ACB, \angle ADB, \angle AEB$ sum to less than $180^{\circ}$, there can only be 1 good circle passing through $A, B$, which is a contradiction. Therefore, $n \geq 6$ is also not possible.

In conclusion, the number of good circles $n = 4$.