
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 $f\left( x \right)=\sin \left( wx-\frac{\pi }{6} \right)+\sin \left( wx-\frac{\pi }{2} \right)$, where $0 < w < 3$, and it is known that $f\left( \frac{\pi }{6} \right)=0$,

$(I)$ Find $w$

$(II)$ Stretch the x-coordinates of the points on the graph of $y=f\left( x \right)$ by a factor of $2$ (the y-coordinates remain unchanged), then shift the graph to the left by $\frac{\pi }{4}$ units to obtain the graph of the function $y=g\left( x \right)$. Find the minimum value of $g\left( x \right)$ on the interval $\left[ -\frac{\pi }{4},\frac{3}{4}\pi \right]$

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