
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 set \( D=\left\{\left(x_{1}, x_{2}\right)\left|x_{1}\right\rangle 0, x_{2}>0, x_{1}+x_{2}=k\right\} \), where \( k \) is a constant, find the range of \( k \) such that the inequality \(\left(\frac{1}{x_{1}}-x_{1}\right)\left(\frac{1}{x_{2}}-x_{2}\right) \geq \left(\frac{k}{2}-\frac{2}{k}\right)^{2}\) holds for any \(\left(x_{1}, x_{2}\right) \in D\).

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