Existence, Uniqueness & Bifurcations · Hard problem

Apply Picard iteration to $y' = 1 + y^2$, $y(0) = 0$: $\varphi_0(t) = 0$ and $\varphi_{n+1}(t) = \displaystyle\int_0^t \big(1 + \varphi_n(s)^2\big)\,ds$. Evaluate $\varphi_3(1)$. Express your answer as a common fraction.