Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

A three-stage explicit Runge–Kutta method uses $k_1 = f(t_n, y_n)$, $k_2 = f\!\left(t_n + \dfrac{h}{4},\ y_n + \dfrac{h}{4}k_1\right)$, $k_3 = f\!\left(t_n + h,\ y_n + h\left(a_{31}k_1 + a_{32}k_2\right)\right)$ with $a_{31} + a_{32} = 1$, and $y_{n+1} = y_n + h(b_1k_1 + b_2k_2 + b_3k_3)$. The coefficients $b_1, b_2, b_3, a_{31}, a_{32}$ are chosen so that the method is third-order accurate: for every smooth $f$, one step agrees with the Taylor expansion of the exact solution through the $h^3$ term. Find $a_{32}$. Express your answer as a common fraction.