
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

An IMO (International Mathematical Olympiad) team leader chooses positive integers \( n \) and \( k \) (where \( n > k \)), and informs the deputy leader and participants of these values. The leader then secretly tells the deputy a binary string consisting of \( n \) digits. The deputy proceeds to write all binary strings of \( n \) digits that have exactly \( k \) digits different from the string given by the leader. For example, if \( n=3 \), \( k=1 \), and the leader chooses 101 to tell the deputy, the deputy should write 001, 111, and 100. The participants are allowed to see the strings written by the deputy and must guess the leader’s string. Determine the minimum number of guesses required (depending on \( n \) and \( k \)) to ensure the correct string is found.

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