
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 $f(x)=A\sin (\omega x+\varphi)$ $(A > 0,\omega > 0,\varphi\in(0, \frac {\pi}{2}))$ has a $y$-intercept of $1$, and takes its maximum and minimum values at the adjacent extremum points $(x_{0}- \frac {3}{2},2)$ and $(x_{0},-2)$ respectively $(x_{0} > 0)$.  
$(1)$ Find the analytic expression of the function $f(x)$;  
$(2)$ If $f(x)= \frac {k+1}{2}$ has two different roots in the interval $[0, \frac {3}{2}]$, find the range of $k$;  
$(3)$ Find the equation of the axis of symmetry of the function $f(x)$ in the interval $[\frac {13}{4}, \frac {23}{4}]$.

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