
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 functions $f(x)$ and $g(x)$ that satisfy the relationship $g(x) = f(x) \cdot f(x + \alpha)$, where $\alpha$ is a constant.
1. If $f(x) = \cos x + \sin x$, and $\alpha = \frac{\pi}{2}$, find the analytical expression for $g(x)$.
2. Design a function $f(x)$ and a value for $\alpha$ such that $g(x) = 2\cos x(\cos x + \sqrt{3}\sin x)$.
3. When $f(x) = |\sin x| + \cos x$, and $\alpha = \frac{\pi}{2}$, there exist $x_1, x_2 \in \mathbb{R}$ such that for any $x \in \mathbb{R}$, $g(x_1) \leq g(x) \leq g(x_2)$ always holds. Find the minimum value of $|x_1 - x_2|$.

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