A twelve versus back-to-back sevens with two dice

Two fair dice are thrown together over and over. Player A wins the moment a total of $12$ is thrown. Player B wins the moment two successive throws both total $7$. Which player is favoured, and what is that player's chance of winning?

A is favoured, winning with probability $7/13$. Track whether the last throw was a $7$. With $a$ and $b$ the chances A wins from the fresh state and the after-a-7 state: $a=\tfrac1{36}+\tfrac16 b+\tfrac{29}{36}a$ and $b=\tfrac1{36}+\tfrac{29}{36}a$, which solve to $a=\tfrac7{13}$.