
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 $f(x)=\sin (\omega x+\frac{π}{3})$, where $\omega\ \ \gt 0$.  
$(1)$ Let $\omega =1$, find the range of $y=f(x)$ for $x\in [0,\pi]$;  
$(2)$ For $a \gt \pi$ ($a\in R$), if the smallest positive period of $f(x)$ is $\pi$, and there are exactly $3$ zeros in $x\in [\pi ,a]$, find the range of $a$.

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