
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 $y=\sin ^{2}x+\sin 2x+3\cos ^{2}x$, find
$(1)$ the minimum value of the function and the set of $x$ values when it occurs;
$(2)$ the interval(s) where the function is strictly decreasing;
$(3)$ the range of $y=f(x)$ when $x\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right]$.

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