
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 $f(x)=\sin (wx- \frac{π}{6})+\sin (wx- \frac{π}{2})$, where $0 < w < 3$. It is known that $f( \frac{π}{6})=0$.

(1) Find the value of $w$;

(2) If the horizontal coordinates of each point on the graph of the function $y=f(x)$ are stretched to twice their original length (the vertical coordinates remain unchanged), and then the resulting graph is shifted to the left by $\frac{π}{4}$ units to obtain the graph of the function $y=g(x)$, find the range of $g(x)$ on $(-\frac{π}{4}, \frac{3π}{4})$.

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