
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
A bag contains 4 identical balls, numbered 0, 1, 2, and 2. Player A draws a ball and puts it back, then player B draws a ball. If the number on the drawn ball is larger, that player wins (if the numbers are the same, it's a tie). What is the probability that player B draws the ball numbered 1, given that player A wins?

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