A sequence of natural numbers is constructed by listing the first $3$, then skipping none, listing the next $4$, skipping $1$, listing $5$, skipping $2$, and on the $n$th iteration, listing $n+2$ and skipping $n-1$. The sequence begins $1,2,3,5,6,7,8,11$. What is the $400,000$th number in the sequence?