
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 $f(x)=2\sin \omega x\cos \omega x+m\ (\omega\ \gt 0, m\in \mathbb{R})$. Choose two out of the three conditions ①, ②, ③ as known to uniquely determine the function $f(x)$.
$(Ⅰ)$ Find the values of $\omega$ and $m$;
$(Ⅱ)$ Let $g(x)=f(x-\frac{π}{6})$, find the maximum value of $g(x)$ on the interval $[0,\frac{π}{2}]$.
Condition ①: $f(0)=1$;
Condition ②: The minimum value of $f(x)$ is $0$;
Condition ③: The distance between two adjacent axes of symmetry of the graph of $f(x)$ is $\frac{π}{2}$.

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