
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 $f(x) = \sin^2\frac{x}{4} - 2\cos^2\frac{x}{4} + \sqrt{3}\sin\frac{x}{4}\cos\frac{x}{4}$, where $x \in \mathbb{R}$. Find the interval where the function $y = f(x)$ is monotonically decreasing. In triangle $ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $f(B) = -\frac{1}{2}$, $b = \sqrt{3}$. Find the range of values for the perimeter of triangle $ABC$.

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