
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 the men's singles final of table tennis at the Tokyo Olympics, Chinese player Ma Long defeated his teammate Fan Zhendong to win the championship. The table tennis final adopts a format of best of $7$ games with a player needing to win $4$ games. In the deciding game, the player who reaches $11$ points first wins, but after reaching a tie at $10$ points, the player must win by a margin of $2$ points. After reaching a tie at $10:10$, the players take turns serving, with each player serving only $1$ ball at a time. If in the deciding game, the probability of Ma Long scoring when he serves is $\frac{2}{3}$, and the probability of Ma Long scoring when Fan Zhendong serves is $\frac{1}{2}$, with each ball's outcome being independent. After the tie at $10:10$, Ma Long serves first. The probability of both players playing until the score reaches $13:11$ is ______.

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