Example 2 In a regular triangular frustum $A B C-A_{1} B_{1} C_{1}$, $A_{1} B_{1}: A B=5: 7$, the section $A B C_{1}$ and $A_{1} B C_{1}$ divide the frustum into three triangular pyramids $C_{1}-A B C$, $C_{1}-A B A_{1}$, and $B-A_{1} B_{1} C_{1}$. Then, the ratio of their volumes $V_{1}$, $V_{2}$, and $V_{3}$ is $\qquad$
Analysis: This problem is the best illustration of Basic Conclusion 4: Let the height of the frustum of a triangular pyramid be $h$, and the areas of the upper and lower bases be $S_{1}$ and $S_{2}$, respectively. Then $V_{1}=\frac{h}{3} S_{2}, V_{2}=\frac{h}{3} \sqrt{S_{1} S_{2}}, V_{3}=\frac{h}{3} S_{1}$, i.e., $V_{1}: V_{2}: V_{3}=S_{2}: \sqrt{S_{1} S_{2}}: S_{1}$. Furthermore, by the similarity of the two bases of the frustum and $A_{1} B_{1}: A B=5: 7$, we know $S_{2}: S_{1}=49: 25$. Therefore, 
$$
V_{1}: V_{2}: V_{3}=S_{2}: \sqrt{S_{1} S_{2}}: S_{1}=49: 35: 25 .
$$