
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 $f(x)=2\cos^2 ωx+2\sqrt{3}\sin ωx\cos ωx+a(ω>0,a∈R)$, and the following conditions:①: The maximum value of $f(x)$ is $1$; ②: A line $x=-\frac{π}{12ω}$ is a symmetry axis of $f(x)$; ③: The distance between two adjacent symmetry axes of $f(x)$ is $\frac{π}{2}$. Choose two reasonable conditions from these three conditions to determine the analytical expression of the function $f(x)$, and find:<br/>$(1)$ The analytical expression of the function $f(x)$;<br/>$(2)$ Given $g(x)=f(2x-\frac{π}{6})$, if the minimum value of $g(x)$ on the interval $\left[0,m\right]$ is $g(0)$, find the maximum value of $m$.

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