Rule of sines: $\frac{b}{\sin \beta}=\frac{a}{\sin 3\beta}=\frac{c}{\sin 4\beta}$, $b=\frac{a}{3-4\sin^{2}\beta}=\frac{c}{4\cos \beta\cos 2\beta}$. First equality: $3b-4b\sin^{2}\beta=a$, giving $\sin^{2}\beta=\frac{3b-a}{4b} \quad (1)$. Second equality: $4b\cos \beta\cos 2\beta=c$ or $4b\cos \beta(1-2\sin^{2}\beta)=c$, using (1), we have $\cos \beta=\frac{c}{2(a-b)}$. Using (1) again, $1-\frac{c^{2}}{4(a-b)^{2}}=\frac{3b-a}{4b}$, Calculating, $(a^{2}-b^{2})(a-b)=bc^{2}$.