
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
Given that \( z \) is a complex number and sets \( A \) and \( B \) are non-empty, if \( A=\left\{z \big| z=\left(2 m-\log _{\sqrt{2}} \frac{k+1}{k}\right)+ \left(m+\log _{\sqrt{2}} \frac{k+1}{k}\right) i\right\} \) and \( B=\{z \big||z| \leqslant 2m-1 \} \), where \( m \) is any real number and \( A \cap B=\varnothing \), then what is the range of the real number \( k \)?

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