
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 \( f(x) = 2x - \frac{2}{x^2} + \frac{a}{x} \), where the constant \( a \in (0, 4] \). Find all real numbers \( k \) such that for \( x_1, x_2 \in \mathbf{R}^+ \), it always holds that \( \left| f(x_1) - f(x_2) \right| > k \left| x_1 - x_2 \right| \).

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