
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 the function $f(x)=2\sin({ωx+\frac{π}{3}})({ω＞0})$ where the distance between two adjacent axes of symmetry on the graph is $\frac{π}{2}$.  
$(1)$ Find the analytical expression and the intervals of monotonicity of $f\left(x\right)$;  
$(2)$ Determine the range of the function $f\left(x\right)$ on the interval $[{-\frac{π}{6},\frac{π}{2}}]$.

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