
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 function $f(x)=\cos \left(x+ \frac {2}{3}\pi\right)+2\cos ^{2} \frac {x}{2}$, where $x\in\mathbb{R}$.  
$(1)$ Find the range of $f(x)$.  
$(2)$ Let $\triangle ABC$ have internal angles $A$, $B$, $C$ and opposite sides of lengths $a$, $b$, $c$ respectively. If $f(B)=1$, $b=1$, and $c= \sqrt {3}$, find the value of $a$.

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