
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\cos 2C+2 \sqrt {2}\cos C+2=0$.  
$(1)$ Find the size of angle $C$;  
$(2)$ If $b= \sqrt {2}a$ and the area of $\triangle ABC$ is $\dfrac { \sqrt {2}}{2}\sin A\sin B$, find the values of $\sin A$ and $c$.

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