
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
Player A and player B are playing a game of Go. If player A plays as black first, the probability of player A winning is $\frac{1}{3}$; if player B plays as black first, the probability of player B winning is $\frac{1}{2}$. Assuming each game is independent and there are no draws, the loser of the first game plays first in the second game. If player A and player B play two games, with the first game having an equal probability of player A or player B playing as black first, then the probability of each player winning one game is ______.

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