
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 the vectors $\overrightarrow{a}=( \sqrt {3}\sin x,-1)$ and $\overrightarrow{b}=(\cos x,m)$ with $m\in \mathbb{R}$:  
$(1)$ If $m= \sqrt {3}$ and $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $\frac {3\sin x-\cos x}{\sin x+\cos x}$;  
$(2)$ Given the function $f(x)=2( \overrightarrow{a}+ \overrightarrow{b})\cdot \overrightarrow{b}-2m^{2}-1$, if $f(x)$ has a zero in the interval $\left[0, \frac {\pi}{2}\right]$, find the range of $m$.

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