
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 $\overrightarrow{a}=(\sin x+\cos x,-2\sin x)$ and $\overrightarrow{b}=(\sin x-\cos x,\sqrt{3}\cos x)$,

(1) Find the value of $x$ in the interval $(0,\pi)$ that satisfies $\overrightarrow{a}\cdot\overrightarrow{b}=2$.

(2) Let $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$. Determine the range of $f(x)$ on the interval $(0,\frac{\pi}{2})$.

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