
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 distance between two adjacent axes of symmetry of the graph of $f(x)$ is $\frac{π}{2}$, please choose any two of conditions ①, ②, ③ as known conditions to answer.<br/>Condition ①: The minimum value of $f(x)$ is $-2$;<br/>Condition ②: The graph of $f(x)$ has a symmetric center at $(\frac{5π}{12},0)$;<br/>Condition ③: The graph of $f(x)$ passes through the point $(\frac{5π}{6},-1)$.<br/>$(1)$ Find the analytical expression of $f(x)$;<br/>$(2)$ In acute triangle $\triangle ABC$, where the internal angles $A$, $B$, $C$ are opposite sides $a$, $b$, $c$ respectively, $A=\frac{π}{3}$, $b=f(A)$, find the range of values for the area of $\triangle ABC$.

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