
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 $f(x)=\sqrt{3}\sin\omega x\cos\omega x+\cos^{2}\omega x-\frac{1}{2}$, where $0 \lt \omega\ \ \lt 2$, with the following three conditions: Condition ①: $f(\frac{π}{3})=\frac{1}{2}$; Condition ②: $f(x+\pi)=f(x)$; Condition ③: $f(\frac{π}{6}-x)=f(\frac{π}{6}+x)$. Choose one of the above three conditions as known and solve the following problems.  
$(1)$ Find the monotonic increasing interval of $f(x)$;  
$(2)$ If the maximum value of $f(x)$ on the interval $[0,m]$ is $1$, find the minimum value of the real number $m$.  
Note: If multiple conditions that meet the requirements are selected for separate answers, the first answer will be scored.

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