
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
Given vectors $\overrightarrow{m}=(\cos \frac {x}{2},-1)$ and $\overrightarrow{n}=( \sqrt {3}\sin \frac {x}{2},\cos ^{2} \frac {x}{2})$, let the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}+1$.  
$(1)$ If $x\in[0, \frac {\pi}{2}]$ and $f(x)= \frac {11}{10}$, find the value of $x$;  
$(2)$ In $\triangle ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and satisfy $2b\cos A\leqslant 2c- \sqrt {3}a$, find the range of $f(B)$.

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