
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 $f(x)=\sin(\omega x-\frac{π}{6})+\sin(\omega x-\frac{π}{2})$, where $0 \lt \omega\ \ \lt 3$, and it is known that $f(\frac{π}{6})=0$.  
$(1)$ Find $\omega$;  
$(2)$ Stretch the abscissa of each point on the graph of the function $y=f\left(x\right)$ by a factor of $2$ (keeping the ordinate unchanged), then shift the resulting graph to the left by $\frac{π}{4}$ units to obtain the graph of the function $y=g\left(x\right)$. Find the interval where $g\left(x\right)$ is monotonically decreasing.

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