
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 function $f(x)=\sin (2x-\frac{π}{3})$, when the graph of function $f(x)$ is shifted to the left by $\frac{π}{4}$ units, the graph of the new function $g(x)$ is obtained.  
$(1)$ Find the expression of function $g(x)$.  
$(2)$ When $x\in [0,\frac{π}{4}]$, the equation $2[g(x)]^{2}-mg(x)+1=0$ has solutions. Find the range of real numbers $m$.  
$(3)$ When $x\in [\frac{π}{6},\frac{π}{3}]$, the inequality $af(x)+g(x)\geqslant \frac{\sqrt{a^2+1}}{2}$ always holds. Find the range of positive number $a$.

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