
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 $f(x) = ax - x^4$, where $x \in \left[\frac{1}{2}, 1\right]$, and $A$, $B$ are two different points on the graph. If the slope $k$ of line $AB$ always satisfies $\frac{1}{2} \leq k \leq 4$, find the value of the real number $a$.

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