
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 $f(x)=A\sin (\omega x+\varphi)$ ($A > 0$, $\omega > 0$, $|\varphi| < \frac {\pi}{2}$, $x\in\mathbb{R}$), and the maximum value of the function $f(x)$ is $2$, the smallest positive period is $\frac {\pi}{2}$, and the graph of the function $f(x)$ passes through the point $(\frac {\pi}{24},0)$.  
$(1)$ Find the analytical expression of the function $f(x)$;  
$(2)$ Let $\triangle ABC$ have angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$, respectively, and $f(\frac {C}{4})=2$, $c= \frac { \sqrt {3}}{2}$, find the range of values for $a+2b$.

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