
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
For the function $f(x)=\sin({ωx+φ})$ ($ω>0$, $|φ|<\frac{π}{2}$), stretch the abscissa of all points on its graph by a factor of $2$, while keeping the ordinate unchanged. Then shift the resulting graph to the left by $\frac{π}{3}$ units to obtain the graph of $g(x)=\sin({2x+\frac{π}{3}})$.  
$(1)$ Find the analytical expression of $f(x)$.  
$(2)$ Determine the range of $f(x)$ on the interval $[0,\frac{π}{3}]$.

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