
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 $f(x)$, $g(x)$ satisfy the relation $g(x)=f(x)\cdot f(x+ \dfrac {π}{2})$,  
$(1)$ If $f(x)=\cos x+\sin x$, find the expression for $g(x)$;  
$(2)$ When $f(x)=|\sin x|+\cos x$, there exist $x_{1}$, $x_{2}\in\mathbb{R}$, such that for any $x\in\mathbb{R}$, $g(x_{1})\leqslant g(x)\leqslant g(x_{2})$ always holds, find the minimum value of $|x_{1}-x_{2}|$.

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