
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
Let the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$ be $a$, $b$, and $c$ respectively, with $b=3$, $c=1$, and $A=2B$.  
$(1)$ Find the value of $a$;  
$(2)$ Find the value of $\sin \left(A+ \frac {\pi}{3}\right)$.

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