Before the game starts, there are a few "points" on the desktop, and then the two players take turns to do the following operations until the operation can not be completed:
Starting from a "point" to draw a line continuously and uninterruptedly, and it does not intersect with any "point" and "line" on the way, and finally ends the curve on a "point". Later, we call the drawn line "line" and choose any point from the line to call it "point" (in fact, which point we choose as "point" will not have any impact).
However, every "point" is required to be connected with at most three "lines". A "line" passes through a "point" means that the "point" is connected with two "lines".
Meanwhile, you can also draw a "line" from a "point" to itself if satisfies the terms above.
My research shows that when the numbers of "points" in the very beginning are $ 1,2,5,6$, there are winning strategies for the second hand. However, when the numbers of "points" are $3,4,7$, there are winning strategies for the first hand. I guess that if the numbers of "points" are in the form of $4n+2, 4n+1$, the second hand has winning strategies.
My question is, is my result right? How to prove it? (And disprove it if it wrong)