
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 $f(x)=\sin(2x-\frac{π}{6})-2\sin^{2}x+1$ for $x\in\mathbb{R}$.  
$(1)$ If $f(\alpha)=\frac{1}{2}$ and $\alpha\in[0,\frac{π}{2}]$, find the angle $\alpha$.  
$(2)$ If the inequality $[f(x)]^{2}+2a\cos(2x+\frac{π}{6})-2a-2＜0$ holds for all $x\in(-\frac{π}{12},\frac{π}{6})$, find the range of real number $a$.  
$(3)$ Shift the graph of function $f(x)$ to the left by $\frac{π}{12}$ units, then keep the ordinate of the points on the graph unchanged and change the abscissa to $\frac{1}{m}$ times the original. This results in the graph of function $g(x)$. If there exists a non-zero constant $\lambda$ such that for any $x\in\mathbb{R}$, $g(x+\lambda)=\lambda g(x)$ holds, find the range of real number $m$.

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