
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 $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system using the same unit of length. The parametric equations of line $l$ are given by $\begin{cases}x=2+t \\ y=1+t\end{cases} (t \text{ is a parameter})$, and the polar coordinate equation of circle $C$ is given by $\rho=4 \sqrt{2}\sin (θ+ \dfrac{π}{4})$.

(1) Find the standard equation of line $l$ and the Cartesian coordinate equation of circle $C$;

(2) Let points $A$ and $B$ be the intersections of curve $C$ and line $l$, and point $P$ has Cartesian coordinates $(2,1)$. Find the value of $||PA|-|PB||$.

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