Example 4 In a square board $ABCD$ with a side length of 8 units, there is a colored equilateral triangle board $OEF$ (vertex $O$ is painted red, $E$ and $F$ are white) placed as shown in Figure 3-2-3 (with $E$ coinciding with $D$, and $EF$ coinciding on the side $AD$). Now, let the triangular color board rotate to the left with $F$ as the pivot until point $O$ touches $AB$, completing one "left advance". Then, with the contact point of $O$ on $AB$ as the pivot, rotate the triangular color board to the left again, making the second "left advance". Continue this process until the 1990th "left advance" and then stop. Ask: How far is the red vertex $O$ of the triangular color board from point $A$ at this time? And explain the reason.
The "left advance" rule for the triangular color board is:
(1) Every 3 left advances, the red vertex $O$ is inside the disk;
(2) Every 16 left advances, the triangular board returns to its original starting position, but the red vertex $O$ moves to point $F$ on the $A D$ edge, and $F$ moves to point $E$.
From this, we can infer that every 48 left advances, the triangular board and its red vertex return to their original positions.
Also, $1990=48 \times 41+22$
Therefore, 1990 left advances are equivalent to 22 left advances. After 16 left advances, $O$ moves to the $F$ position. After another 6 left advances, this point moves to the $A B$ edge 1. Let's assume this point is $P$, at this time $F$ is exactly at the midpoint of $A D$, so $P A=3$.

Explanation: Here, we use the status after 16 left advances to deduce a cycle of 48. Without the help of this periodicity, we would be stuck in mechanical repetitive trials.

Many geometric problems, through exploration and discovery, can reveal special patterns, one of which is the periodic phenomenon.