
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 $\overrightarrow{a}=({2\cos x, \cos x - \sqrt{3}\sin x}), \overrightarrow{b}=({\sin(x+\frac{\pi}{3}), \sin x})$, and the function $f(x)=\overrightarrow{a} \cdot \overrightarrow{b}$.  
$(1)$ Find the smallest positive period of the function $y=f(x)$.  
$(2)$ Determine the intervals where the function $y=f(x)$ is decreasing.  
$(3)$ If the function $y=f(x+\varphi)$ (where $\varphi \in [0,\pi]$) is an even function on $R$, find the value of $\varphi$.

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