
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 $f(x)=Asin(ωx+φ)(A>0,ω>0,|φ|<\frac{π}{2})$ and the graph of $f(x)$ passes through $A(\frac{π}{3},2)$ and $B(-\frac{π}{6},-2)$ within one period, and the graph of $f(x)$ is symmetric about the line $x=\frac{π}{3}$. <br/>$(1)$ Find the analytical expression of $f(x)$; <br/>$(2)$ If there exists $x∈[\frac{3π}{4},\frac{7π}{6}]$ such that the inequality $f(x)≤2a+3$ holds, find the range of values for $a$.

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