
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 and B take turns shooting baskets, each shooting once per turn. It is agreed that A shoots first and the first to make a shot wins. The game ends when someone wins or when both have shot three times. Suppose the probability of A making a shot each time is $\frac{1}{3}$, and the probability of B making a shot each time is $\frac{1}{2}$, with each shot being independent of the others.  
(Ⅰ) Find the probability of B winning;  
(Ⅱ) Find the probability that the game ends with B having shot only 2 balls.

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