
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 $f(x)=\sin (\omega x- \frac {\pi}{6})+\sin (\omega x- \frac {\pi}{2})$, where $0 < \omega < 3$, and it is known that $f( \frac {\pi}{6})=0$.  
(I) Find $\omega$;  
(II) Stretch the x-coordinates of the points on the graph of the function $y=f(x)$ by a factor of $2$ (keeping the y-coordinates unchanged), then shift the resulting graph to the left by $\frac {\pi}{4}$ units to obtain the graph of the function $y=g(x)$. Find the minimum value of $g(x)$ on the interval $[- \frac {\pi}{4}, \frac {3\pi}{4}]$.

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