
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 $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and the sizes of angles $A$, $B$, $C$ form an arithmetic sequence. Let vector $\overrightarrow{m}=(\sin \frac {A}{2},\cos \frac {A}{2})$, $\overrightarrow{n}=(\cos \frac {A}{2},- \sqrt {3}\cos \frac {A}{2})$, and $f(A)= \overrightarrow{m} \cdot \overrightarrow{n}$,  
$(1)$ If $f(A)=- \frac { \sqrt {3}}{2}$, determine the shape of $\triangle ABC$;  
$(2)$ If $b= \sqrt {3}$ and $a= \sqrt {2}$, find the length of side $c$ and the area $S_{\triangle ABC}$.

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