
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 function \( f(x) = \frac{1}{x} - x \). If for any \( x \in (0, k) \), it holds that \( f(x) \cdot f(k - x) \geq \left(\frac{k}{2} - \frac{2}{k}\right)^2 \). Find the range of values for the positive real number \( k \).

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