
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 $f(x)=\sin^2x+ \sqrt{3}\sin x\cdot\cos x+2\cos^2x$, where $x\in\mathbb{R}$  
$(1)$ Find the smallest positive period and the intervals of monotonic decrease for the function $f(x)$.  
$(2)$ How can the graph of the function $f(x)$ be obtained by transforming the graph of the function $y=\sin 2x$?

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