Conditional Probability, Independence & Bayes · Challenge problem

Teams $A$ and $B$ play a best-of-seven series: the first team to win $4$ games wins the series, and no further games are played. Team $A$ wins each game with probability $\dfrac23$, independently of the other games. Given that team $A$ wins the series, what is the probability that the series lasted all seven games? Express your answer as a common fraction.