Existence, Uniqueness & Bifurcations · Superchallenge problem

Apply Picard iteration to $y' = y^2$, $y(0) = 1$: $\varphi_0(t) = 1$ and $\varphi_{n+1}(t) = 1 + \displaystyle\int_0^t \varphi_n(s)^2\,ds$. Each $\varphi_n$ is a polynomial that agrees with the exact solution $\dfrac{1}{1 - t} = 1 + t + t^2 + \cdots$ through the $t^n$ term. Find the coefficient of $t^8$ in $\varphi_7(t)$. Express your answer as a common fraction.