Define a regular \(n\)-pointed star to be the union of \(n\) line segments \(P_1P_2, P_2P_3,\ldots, P_nP_1\) such that
the points \(P_1, P_2,\ldots, P_n\) are coplanar and no three of them are collinear,
each of the \(n\) line segments intersects at least one of the other line segments at a point other than an endpoint,
all of the angles at \(P_1, P_2,\ldots, P_n\) are congruent,
all of the \(n\) line segments \(P_2P_3,\ldots, P_nP_1\) are congruent, and
the path \(P_1P_2, P_2P_3,\ldots, P_nP_1\) turns counterclockwise at an angle of less than 180 degrees at each vertex.
There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?
(第二十二届AIME1 2004 第8题)