(Method One)
As shown in the figure, draw the ball's trajectory according to the requirements. The ball reaches point $C$ for the first time after 14 bounces.
(Method Two)
Fold the rectangle $A B C D$ along $B C$;
The path of the ball from $A \rightarrow E \rightarrow F$ is the same as from $A \rightarrow E \rightarrow F^{\prime}$.
When the ball returns to one of the vertices of the rectangle again, the shape formed by the original rectangle and the folded part is a square;
$$
[9,7]=9 \times 7=63 \text {; }
$$

63 rectangles with a length of 7 and a width of 9 can form a square with a side length of 63; the number of collisions the ball experiences is the same as the number of times it crosses a fold line;

The diagonal of the square crosses $(63 \div 7-1)+(63 \div 9-1)=14$ fold lines;
Therefore, when the ball returns to one of the vertices of the rectangle again, it has experienced 14 collisions.
As shown in the figure, on a rectangular table surrounded by four sides, with a length of $9 \mathrm{~cm}$ and a width of $7 \mathrm{~cm}$, a small ball is shot out from point $A$ at a 45-degree angle, reaches point $E$ and bounces back at a 45-degree angle, and continues to roll forward. During the entire process, the ball bounces back at a 45-degree angle each time it hits the edge of the table. Try to answer, after leaving point $A$, how many times does the ball bounce before it first reaches point $C$?