
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
The function $f(x)=2\sin (\omega x- \frac {\pi}{6})+1(A > 0,\omega > 0)$, the distance between two adjacent axes of symmetry of its graph is $\frac {\pi}{2}$  
$(1)$ Find the analytical expression of the function $f(x)$;  
$(2)$ When $x\in[0, \frac {\pi}{2}]$, find the maximum value of $f(x)$.

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