
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 functions
\[ f(x) = x + \frac{a}{x} - 4 \]
\[ g(x) = kx + 3 \]
For \( a \in [1, 2] \), if the inequality 
\[ \left| f(x_1) \right| - \left| f(x_2) \right| < g(x_1) - g(x_2) \]
holds for any \( x_1, x_2 \in [2, 4] \) with \( x_1 < x_2 \) always, determine the range of the real number \( k \).

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