
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 vectors $\overrightarrow{a}=(1,\sin x)$ and $\overrightarrow{b}=(\cos(2x + \frac{\pi}{3}), \sin x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} - \frac{1}{2}\cos 2x$.

(1) Find the explicit expression of the function $f(x)$ and its intervals of monotonic increase.

(2) In $\triangle ABC$, if the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively, given that $c = \sqrt{3}$ and $f(C) = 0$, find the range of possible values for the perimeter of $\triangle ABC$.

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