
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
Let the function $f(x) = 2\sqrt{3}\sin(2\omega x + \frac{\pi}{3}) - 4\cos^2\omega x + 3$ ($0 < \omega < 2$), and the graph of $y = f(x)$ has a symmetry axis at $x = \frac{\pi}{6}$.  
(1) Find the value of $\omega$ and the minimum value of $f(x)$.  
(2) In $\triangle ABC$, $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively, with $a=1$ and the area of $\triangle ABC$, $S_{\triangle ABC} = \frac{\sqrt{3}}{4}$. Given $f(A) = 2$, find the perimeter of $\triangle ABC$.

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