
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 the equation \( |x-2n| = k \sqrt{x} \) (where \( n \in \mathbf{N} \)), determine the range of values for \( k \) such that the equation has two distinct real roots in the interval \( (2n-1, 2n+1] \).

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