
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 a function defined on $\mathbb{R}$, $f(x)=A\sin (\omega x+\varphi)$ where $A > 0$, $\omega > 0$, and $|\varphi| \leqslant \frac {\pi}{2}$, the minimum value of the function is $-2$, and the distance between two adjacent axes of symmetry is $\frac {\pi}{2}$. After the graph of the function is shifted to the left by $\frac {\pi}{12}$ units, the resulting graph corresponds to an even function.  
$(1)$ Find the expression for the function $f(x)$.  
$(2)$ If $f\left( \frac {x_{0}}{2}\right)=- \frac {3}{8}$, and $x_{0}\in\left[ \frac {\pi}{2},\pi\right]$, find the value of $\cos \left(x_{0}+ \frac {\pi}{6}\right)$.

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