
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 the Cartesian coordinate system $xOy$ with the origin as the pole and the positive half-axis of $x$ as the polar axis, establish a polar coordinate system. Both coordinate systems use the same unit of length. The parametric equation of line $l$ is $\begin{cases} x=2+ \dfrac { \sqrt {2}}{2}t \\ y=1+ \dfrac { \sqrt {2}}{2}t \end{cases}$, and the polar equation of circle $C$ is $\rho=4 \sqrt {2}\sin (\theta+ \dfrac {\pi}{4}).$  
$(1)$ Find the standard equation of line $l$ and the Cartesian coordinate equation of circle $C$;  
$(2)$ Suppose curve $C$ intersects line $l$ at points $A$ and $B$. If point $P$ has Cartesian coordinates $(2,1)$, find the value of $||PA|-|PB||.$

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