
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

Two people, A and B, are playing table tennis. The probability that A wins a game is \(\frac{2}{3}\). The match ends when one of the players is ahead by two games. The probability that the total number of games is \(x_i \left(i \in \mathbf{N}_{+}\right)\) is \(p_i\), where \(x_i < x_{i+1} \left(i \in \mathbf{N}_{+}\right)\). Find \(S=\sum_{i=1}^{+\infty} x_{i} p_{i}\).

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