
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 $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, establish a polar coordinate system. Given that the parametric equations of line $l$ are $\begin{cases}x=-1+t \\ y=1+t\end{cases}$ (with $t$ as the parameter), and the general equation of curve $C$ is $(x-2)^2+(y-1)^2=5$. The polar coordinates of point $P$ are $(2\sqrt{2}, \frac{7\pi}{4})$.

$(I)$ Find the general equation of line $l$ and the polar equation of curve $C$.

$(II)$ If line $l$ is translated 2 units to the right to obtain line $l'$, assume $l'$ intersects with $C$ at points $A$ and $B$. Find the area of triangle $PAB$.

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