
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 $\overrightarrow{a}=(2\sin x,\cos ^{2}x)$, $\overrightarrow{b}=(\sqrt{3}\cos x,2)$, $f\left(x\right)=\overrightarrow{a}\cdot \overrightarrow{b}$.  
$(1)$ Find the smallest positive period and the interval of monotonic decrease of $f\left(x\right)$;  
$(2)$ Find the maximum and minimum values of the function $f\left(x\right)$ on the interval $[0,\frac{π}{2}]$.

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