
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
Given that the probability of player A winning each game is $p$, the probability of player B winning each game is $1-p$. The game follows a best-of-three format, where each game must have a winner and there are no ties. Let random variable $X$ represent the total number of games played. Given that the expected value of $X$ is $\frac{22}{9}$, calculate $p$.

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