
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 function $f(x)=\sin (2x- \dfrac {\pi}{4})+\cos (2x- \dfrac {3\pi}{4})$, where $x\in\mathbb{R}$.  
$(1)$ Find the smallest positive period and the maximum value of $f(x)$;  
$(2)$ Given $\alpha$, $\beta\in(0, \dfrac {\pi}{2})$, and $f(\alpha)= \sqrt {2}$, $\cos (\alpha+\beta)= \dfrac {1}{3}$, find the value of $\tan \beta$.

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