
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 line $l$ is $\begin{cases} x=- \dfrac { \sqrt {3}}{2}t \\ y=1+ \dfrac {1}{2}t\end{cases}$ (where $t$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinate system is established. The equation of curve $C_{1}$ is $\rho= \dfrac {36}{4 \sqrt {3}\sin \theta-12\cos \theta-\rho}$. Let $M(6,0)$ be a fixed point, and $N$ be a moving point on curve $C_{1}$, with $Q$ being the midpoint of $MN$.  
$(1)$ Find the Cartesian coordinate equation of the trajectory $C_{2}$ of point $Q$;  
$(2)$ Given that the intersection of line $l$ with the $x$-axis is point $P$, and its intersections with curve $C_{2}$ are points $A$ and $B$. If the midpoint of $AB$ is $D$, find the length of $|PD|$.

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