Gauss-Manin connection of a family over a formal disk

Question

Let $X$ be a smooth proper scheme over $\mathbb{C}[[t]]$ and $X_0$ its special fiber. The cohomology $H^*_{dR} (X/\mathbb{C}[[t]])$ admit the Gauss-Manin connection which is determined by the action of $\nabla_{\partial_t}$. Firstly, how to compute horizontal sections of this connection? Secondly, one paper says that there is a map $H^*_{dR} (X/\mathbb{C}[[t]]) \to (H^*_{dR} (X/\mathbb{C}[[t]]))^{\nabla=0}$ given by $\pi := \sum ((-t)^i/i!) \nabla_{\partial_t}^i$. I don't see that; I want to check that $\nabla_{\partial_t} (\pi(x)) = 0$ for any cohomology class $x$. But $\nabla_{\partial_t} (\pi(x)) = \sum ((-t)^i/i!) \nabla_{\partial_t}^{i+1}$ and there's no reason why it's zero so probably I'm misunderstanding something important.