
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
Consider the function $f(x) = \sin (\omega x + \varphi)$ with $\omega > 0$ and $|\varphi| < \frac{\pi}{2}$. It is given that the terminal side of the angle $\varphi$ passes through the point $(1, -\sqrt{3})$. Let $M(x_1, y_1)$ and $N(x_2, y_2)$ be any two points on the graph of $f(x)$. When $|f(x_1) - f(x_2)| = 2$, the minimum value of $|x_1 - x_2|$ is $\frac{\pi}{2}$.
1. Determine the explicit formula for the function $y=f(x)$.
2. Given that the area of triangle $ABC$ is $5\sqrt{3}$ and the side $c$ opposite angle $C$ equals $2\sqrt{5}$, with $\cos C = f(\frac{\pi}{4})$, find the perimeter of triangle $ABC$.

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