
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 $f(x)=2\sin x\cos x+\sin ^{2}x-\cos ^{2}x.$  
$(1)$ Find the interval of monotonic decrease for the function $f(x)$;  
$(2)$ If the graph of $f(x)$ is translated to the left by $\dfrac {\pi}{8}$ units and then the x-coordinates are shortened to half of their original length without changing the y-coordinates, the graph of function $g(x)$ is obtained. Find the axis of symmetry for $g(x)$;  
$(3)$ If $f(- \dfrac {\alpha}{2})=- \dfrac { \sqrt {3}}{3}$, $(\alpha\in(0,\pi))$, find the value of $\cos 2\alpha.$

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