The Gambler's Friend

If one die is chosen from a bag of six dice each given a number from $1$ to $6$ such that the $n^{th}$ die has a $\frac{2}{7}$ chance of rolling $n$ and a $\frac{1}{7}$ chance of rolling any other of the numbers from $1$ to $6$ . Given that the die chosen rolled the sequence $(1,1,2,3,4,5,6)$ $k$ times and the probability that it is the first die is $\frac{1}{2}$ or greater what is the minimum value of $k$