
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 $f(x)=2\sin x(\sin x+\cos x)$, $x\in R$.  
$(1)$ Find the smallest positive period and the interval of monotonic increase of $f(x)$;  
$(2)$ The graph of $f(x)$ is shifted to the left by $\frac{π}{4}$ units to obtain the graph of $g(x)$. Find the range of $y=g(x)$, $x∈[0,\frac{π}{2}]$.

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