
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 \(n \in \mathbf{N}\)) has two distinct real roots in the interval \((2n-1, 2n+1]\), determine the range of values for \(k\).

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