
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 $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and it satisfies $\frac {2a-b}{c}= \frac {\cos B}{\cos C}$,  
$(1)$ Find the size of angle $C$;  
$(2)$ Let the function $f(x)=2\sin x\cos x\cos C+2\sin ^{2}x\sin C- \frac { \sqrt {3}}{2}$, find the range of the function $f(x)$ in the interval $\left[0, \frac {\pi}{2}\right]$.

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