
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 $f(x) = \left| \frac{4}{x} - ax \right|$, if for any positive real number $a$, there always exists $x\_0 \in [1, 4]$ such that $f(x\_0) \geq m$, determine the range of the real number $m$.

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