
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
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_{1}$ is $\begin{cases}x= \frac{1}{2}t \\ y=m+ \frac{ \sqrt{3}}{2}t\end{cases} (t$ is the parameter$)$, in the polar coordinate system with the origin $O$ as the pole and the positive direction of the $x$-axis as the polar axis, the polar equation of curve $C_{2}$ is $p=4\cos \left(\theta- \frac{\pi}{6}\right)$.  
$(1)$ Write the Cartesian coordinate equation of curve $C_{2}$;  

$(2)$ Suppose points $P$, $Q$ are moving on $C_{1}$, $C_{2}$ respectively, if the minimum value of $|PQ|$ is $1$, find the value of $m$.

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