
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 $f(x)={\sin^2}ωx+\sqrt{3}\sinωx\cosωx$ where $\omega\gt 0$, the distance between two adjacent symmetric axes of the graph is $\frac{\pi}{2}$.  
$(Ⅰ)$ Find the value of $\omega$;  
$(Ⅱ)$ Determine the intervals where the function $f(x)$ is decreasing;  
$(Ⅲ)$ If for any $x_1, x_2 \in [0, \frac{\pi}{2}]$ we have $|f(x_{1})-f(x_{2})| \lt m$, find the range of real numbers for $m$.

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