
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
The Brazilian IMO leader chooses two natural numbers \( n \) and \( k \) with \( n > k \), and tells these numbers to his Deputy and a participant. Then, the leader whispers a binary sequence of length \( n \) into the Deputy's ear. The Deputy writes down all binary sequences of length \( n \) that differ from the leader's sequence at exactly \( k \) positions. (Example for \( n = 3 \) and \( k = 1 \): If the leader chooses 101, the Deputy writes down 001, 100, 111.) The participant looks at the sequences written down by the Deputy. The participant then tries to determine the original sequence chosen by the leader.

How many times must the participant guess (depending on \( n \) and \( k \)) to ensure that they have guessed correctly at least once?

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