
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
In the trigonometric function section, we have studied the double angle formula $\cos 2x=2\cos ^{2}x-1$. We can continue in a similar way to obtain the triple angle formula. Based on your research results, solve the following problem: In the acute triangle $\triangle ABC$, where the opposite sides of angles $A$, $B$, $C$ are $a$, $b$, $c$, if $A≤\frac{π}{3}$, $\cos C+4\cos ^{3}A-3\cos A=0$, then the range of $4\tan A+\frac{1}{\tan(B-A)}$ is ______.

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