Numerical Methods: Euler to Runge–Kutta · Challenge problem

The boundary value problem $y'' = -y$, $y(0) = 0$, $y(1) = 1$ is solved by shooting: write it as $u' = v$, $v' = -u$ with $u(0) = 0$, $v(0) = s$, march to $t = 1$ with two steps of Heun's method ($h = \dfrac12$; $\mathbf{k}_1 = F(\mathbf{w}_n)$, $\mathbf{k}_2 = F(\mathbf{w}_n + h\mathbf{k}_1)$, $\mathbf{w}_{n+1} = \mathbf{w}_n + \dfrac{h}{2}(\mathbf{k}_1 + \mathbf{k}_2)$), and choose the slope $s$ so that the computed $u$ at $t = 1$ equals $1$. Find $s$. Express your answer as a common fraction.