Increasing Chains

Question

Fix positive integers $n$ and $k$, and suppose $X_1,\dots,X_n \sim \text{Unif}(0,1)$ IID. We say that there is an $\textit{increasing k-chain}$ starting from position $i$ if $X_i < X_{i+1} < \dots < X_{i + (k-1)}$. Find $n$ such that the expected number of increasing $6-$chains among $X_1,\dots,X_n$ is $1$.