SEAMO 2027 - Paper F - SET 06 - Self-study Question 10

The side lengths of a triangle are successive terms of a geometric progression. Let $A$ and $C$ be its smallest and largest interior angles, respectively. Its shortest side has length 16 cm and
$$\frac{\sin A-2\sin B+3\sin C}{\sin C-2\sin B+3\sin A}=\frac{19}{9}.$$
Find its perimeter in centimetres.