5. (3 points) In a regular triangular pyramid (not a regular tetrahedron), the area of the base is three times smaller than the area of a lateral face. The height of the pyramid is 10 cm.

Let's construct the following (infinite) sequence of spheres. Let \( S_{1} \) be the inscribed sphere of this pyramid. Then \( S_{2} \) is the sphere that touches the lateral faces of the pyramid and the sphere \( S_{1} \); \( S_{3} \) is the sphere that touches the lateral faces of the pyramid and the sphere \( S_{2} \), not equal to \( S_{1} \), and so on; \( S_{n+1} \) is the sphere that touches the lateral faces of the pyramid and the sphere \( S_{n} \), not equal to \( S_{n-1} \).

Find the total volume of all these spheres.
Answer: $\frac{500}{183} \pi \mathrm{cm}^{3}$.