
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
There are many two kinds of black and white cards. When you have  $ k$  cards in hand, consider the following procedure  $ (\bf{A})$ .

 $ (\bf{A})$  You choose one card from  $ k$  cards in hand with equal probability  $ \frac {1}{k}$  and replace the card with different color one.

Answer the following questions.

(1) When you have 4 cards with 2 white and 2 black, first,  find the probabilty such that  you will first have 4 cards in same color after repeating the procedure  $ \bf{(A)}$ 
 $ n$  times.

(2) When you have 6 cards with 3 white and 3 black, first,  find the probabilty such that  you will first have 6 cards in same color after repeating the procedure  $ \bf{(A)}$  
 $ n$  times.

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