
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 vectors $a( \sqrt {3}\cos \omega x,\sin \omega x)$ and $b(\sin \omega x,0)$, with $\omega > 0$, let the function $f(x)=(a+b)\cdot b+k$.  
$(1)$ If the distance between two adjacent axes of symmetry in the graph of $f(x)$ is not less than $\dfrac {\pi}{2}$, find the range of $\omega$.  
$(2)$ If the smallest positive period of $f(x)$ is $\pi$, and when $x\in[- \dfrac {\pi}{6}, \dfrac {\pi}{6}]$, the maximum value of $f(x)$ is $2$, find the value of $k$.

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