
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
Let \( p \) and \( q \) be positive real numbers with \( p + q < 1 \). Teams \( A \) and \( B \) play a series of games. For each game, \( A \) wins with probability \( p \), \( B \) wins with probability \( q \), and they tie with probability \( 1-p-q \). The series ends when one team has won two more games than the other, that team being declared the winner of the series. What is the probability that \( A \) wins the series?

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