
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 plane rectangular 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 with the same unit of length. The parametric equation of line $l$ is $\begin{cases}x=2+\frac{\sqrt{2}}{2}t\\y=1+\frac{\sqrt{2}}{2}t\end{cases}$, and the polar coordinate equation of circle $C$ is $\rho=4\sqrt{2}\sin\left(\theta+\frac{\pi}{4}\right)$.

(1) Find the ordinary equation of line $l$ and the rectangular coordinate equation of circle $C$.
(2) Suppose curve $C$ intersects with line $l$ at points $A$ and $B$. If the rectangular coordinate of point $P$ is $(2,1)$, find the value of $||PA|-|PB||$.

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