When the Stopping Rule Looks at the Last Draw

Draw i.i.d. values $X_1,X_2,\dots$ uniform on $\{1,2,3\}$ (so $E[X_i]=2$). Define $N$ as follows: keep drawing and stop the first time you draw a $3$; let $N$ be the number of draws. Let $S_N=\sum_{i=1}^N X_i$. A candidate computes $E[N]E[X_1]=3\cdot 2=6$ and claims $E[S_N]=6$. Compute the correct value of $E[S_N]$ and explain in one sentence why $E[N]E[X_1]$ is the wrong formula here.