
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 $\overrightarrow{a}=(\sqrt{3}\sin x,1)$, $\overrightarrow{b}=(1,-\cos x)$, where $x\in \mathbb{R}$.  
$(1)$ If $x=0$, find $\overrightarrow{a} \cdot \overrightarrow{b}$;  
$(2)$ Let $f(x)=\overrightarrow{a} \cdot \overrightarrow{b}$, find the interval where $f(x)$ is monotonically increasing.

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