Proof that two conditional expectations are equal

Question

Let $(X_n)_{n \geq 1}$ be a sequence of independent and identically distributed random variables in $L^{1}(\Omega,\mathcal{F}, P )$. If $S_n = \sum_{i=1}^{n}X_{i}$ and $\mathcal{G}_{n} = \sigma (S_{n},S_{n+1}, \dots)$, I need to show $$ E(X_{1} \mid \mathcal{G}_{n} ) = E(X_{1} \mid S_{n} ). $$ I suspect $\mathcal{G}_{n} \subset S_{n}$ for if $B = B_{n} \times B_{n+1} \times \cdots ∈ \mathcal{B}(\mathbb{R} \times \mathbb{R} \times \cdots)$, $$(S_{n},S_{n+1},\dots)^{-1}(B)= \{ S_{n} ∈ B_{n}, S_{n+1} ∈ B_{n+1}, \dots \} = \bigcap_{k\geq 0} \{ S_{n+k} ∈ B_{n+k}\} \subset \{ S_{n} ∈ B_n\}$$ I don't know where to go from there. I'd appreciate a hint so I can proceed.