
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 two independent events $A$ and $B$, the probability that neither of them occurs is $\frac{1}{9}$. If the probability that event $A$ occurs and event $B$ does not occur is equal to the probability that event $B$ occurs and event $A$ does not occur, calculate the probability $P(A)$ that event $A$ occurs.

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