
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 $f(x)=2\sin(\omega x+\frac{\pi}{3})+m-\sqrt{3}(\omega>0)$. Among the three conditions ①, ②, and ③, choose two conditions that can determine the values of $\omega$ and $m$ as known. Condition ①: $f(0)=2$; Condition ②: the sum of the maximum and minimum values of $f(x)$ is $0$; Condition ③: the minimum positive period of $f(x)$ is $\pi$.  
$(1)$ Find the value of $f(\frac{\pi}{6})$;  
$(2)$ If the function $f(x)$ is increasing on the interval $[0,a]$, find the maximum value of the real number $a$.

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