
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In a table tennis match, each game is played in an $11$-point system, where the player who reaches $11$ points first wins. However, if the score reaches a tie at $10:10$, the winner must be the first player to gain a lead of $2$ points. After reaching a tie at $10:10$, the players alternate serving, with each player serving only one ball at a time. In a match between players A and B, when player A serves, the probability of A scoring is $\frac{2}{3}$, and when player B serves, the probability of A scoring is $\frac{1}{2}$. The outcomes of each ball are independent. If the score is tied at $10:10$ and player A serves first, the probability of both players playing until a score of $13:11$ is ______.

Present the answer in LaTex format: \boxed{Your answer}
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