
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
In-Young generates a string of \( B \) zeroes and ones using the following method:
- First, she flips a fair coin. If it lands heads, her first digit will be a 0, and if it lands tails, her first digit will be a 1.
- For each subsequent bit, she flips an unfair coin, which lands heads with probability \( A \). If the coin lands heads, she writes down the number (zero or one) different from the previous digit, while if the coin lands tails, she writes down the previous digit again.
What is the expected value of the number of zeroes in her string?

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