
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 $\overrightarrow{a}=(\sqrt{3},1)$, $\overrightarrow{b}=(\sin x,-\cos x)$ $(x\in\mathbb{R})$.
$(1)$ If $\overrightarrow{a}\parallel \overrightarrow{b}$, and $x\in[0,\pi]$, find the value of $x$;
$(2)$ Let the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, the graph of function $f(x)$ is translated left by $\frac{\pi}{3}$ units to obtain the graph of function $g(x)$. When $x\in[0,\pi]$, find the range of function $g(x)$.

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