
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 $f(x)=\sin (\omega x-\varphi)$ $(\omega > 0,|\varphi| < \frac {\pi}{2})$ whose graph intersects with the x-axis at points that are a distance of $\frac {\pi}{2}$ apart, and it passes through the point $(0,- \frac {1}{2})$  
$(1)$ Find the analytical expression of the function $f(x)$;  
$(2)$ Let the function $g(x)=f(x)+2\cos ^{2}x$, find the maximum and minimum values of $g(x)$ in the interval $\left[0, \frac {\pi}{2}\right]$.

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