
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 acute triangle $ \Delta ABC $, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $\tan A - \tan B = \frac{\sqrt{3}}{3}(1 + \tan A \tan B)$.

$(1)$ If $c^2 = a^2 + b^2 - ab$, find the magnitudes of angles $A$, $B$, and $C$.

$(2)$ Given vectors $\vec{m} = (\sin A, \cos A)$, $\vec{n} = (\cos B, \sin B)$, find the range of values for $|3\vec{m} - 2\vec{n}|$.

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