Calculating the constants for Runge-Kutta order 4 in other form

Question

I know why Runge-Kutta order 4 can be written in the below form I guess. But I don't know how I should go about to calculate the constants required.
Runge-Kutta order 4 can also be written in the below form: $$w_0 = \alpha_{0}$$ $$ w_{i+1} = w_i \begin{aligned}[t]&+\frac{h}{6} f(t_i, w_i) \\&+\frac{h}{3}f(t_i+\alpha_1h,w_i+\delta_1f(t_i,w_i)) \\&+\frac{h}{3}f(t_i+\alpha_2h,w_i+\delta_2hf(t_i+\gamma_2h,w_i+\gamma_3hf(t_i,w_i))) \\&+\frac{h}{6}f(t_i+\alpha_3h,w_i+\delta_3hf(t_i+\gamma_4h,w_i+\gamma_5hf(t_i+\gamma_6h,w_i+\gamma_7hf(t_i,w_i)))) \end{aligned}$$ How should I find out the constants $\alpha_1$, $\alpha_2$, $\alpha_3$, $\delta_1$, $\delta_2$, $\delta_3$, $\gamma_2$, $\gamma_3$, $\gamma_4$, $\gamma_5$, $\gamma_6$, $\gamma_7$ ?
I tried to somehow write the Taylor expansion for the function and compare the coefficients but my solution led to some unsolvable equations.