
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, a polar coordinate system is established. It is known that the parametric equation of line $l$ is $$\begin{cases} x=t \\ y=t+2 \end{cases}$$ (where $t$ is the parameter), and the standard equation of curve $C$ is $x^2-4x+y^2-2y=0$. The polar coordinates of point $P$ are $(2\sqrt{2}, \frac{7\pi}{4})$.
(1) Find the standard equation of line $l$ and the polar equation of curve $C$;
(2) If line $l$ is translated 2 units to the right to obtain line $l'$, and $l'$ intersects $C$ at points $A$ and $B$, find the area of $\triangle PAB$.

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