
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 $f(x)=A\sin (\omega x+\varphi)$ where $A > 0$, $\omega > 0$, $-\pi < \varphi \leqslant \pi$ reach its maximum value of $2$ at $x= \frac {\pi}{6}$, and the distance between two adjacent intersections of its graph with the $x$-axis is $\frac {\pi}{2}$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ Find the range of the function $g(x)= \frac {6\cos ^{4}x-\sin ^{2}x-1}{[f( \frac {x}{2}+ \frac {\pi}{6})]^{2}-2}$.

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