
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 $f(x)=\sin^2x+2\sin x\cos x+3\cos^2x$, where $x\in\mathbb{R}$. Find:  
$(1)$ The minimum value of the function $f(x)$ and the set of values of the variable $x$ at which the minimum value is obtained;  
$(2)$ The interval(s) where the function $f(x)$ is decreasing.

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