
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 $\overrightarrow{m}=(2\cos \omega x,-1)$ and $\overrightarrow{n}=( \sqrt {3}\sin \omega x+\cos \omega x,1)$ ($\omega > 0$), and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$. If the graph of the function $f(x)$ intersects the $x$-axis at two adjacent points with a distance of $\dfrac {\pi}{2}$.  
$(1)$ Find the range of the function $f(x)$ on $[0, \dfrac {\pi}{2}]$;  
$(2)$ In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $f(A)=1$, $a=3$, and the height from $BC$ is $\dfrac {3 \sqrt {3}}{2}$, find the values of $b$ and $c$.

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