
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 $\overrightarrow{a} = (3\sin x, -1)$ and $\overrightarrow{b} = (3\cos x, 2)$, where $x \in \mathbb{R}$.

(1) If $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $\sin 2x$;

(2) Let vector $\overrightarrow{c} = \left(\frac{1}{3}, -\frac{1}{3}\right)$, and define $f(x) = \frac{1}{9}(\overrightarrow{a} + \overrightarrow{b}) \cdot (\overrightarrow{a} - \overrightarrow{b}) + \overrightarrow{a} \cdot \overrightarrow{c}$, where $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, find the range of the function $f(x)$.

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