
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 functions $f(x)$ and $g(x)$ satisfy the relationship $$g(x)=f(x)\cdot f\left(x+ \frac {\pi}{2}\right),$$  
(1) If $f(x)=\cos x+\sin x$, find the explicit formula for $g(x)$;  
(2) When $f(x)=|\sin x|+\cos x$, 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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