
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 the function $f(x)=\sin 2x+2\cos ^{2}x-1$.  
$(1)$ Find the smallest positive period of $f(x)$;  
$(2)$ When $x∈[0,\frac{π}{2}]$, find the minimum value of $f(x)$ and the corresponding value of the independent variable $x$.

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