Spinner Duel

You and a rival trader settle a bet with a wheel divided into 10 equal sectors, numbered 1 through 10. Every spin lands on each sector with probability $1/10$, independently. You take turns spinning, and you go first.

• On your turn, if the wheel lands on 1, 2, or 3, you win and the game ends.
• If you did not win, your rival spins. If the wheel lands on 5, 6, 7, 8, 9, or 10, she wins and the game ends.
• If neither spin produced a winner, the game continues in the same way (you spin again, and so on) until someone wins.

Let $p_1$ be the probability that you win and $p_2$ the probability that your rival wins. Find $\max\{p_1,p_2\}$ as an exact fraction.