
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 \(xOy\), an alternative polar coordinate system is established with the origin \(O\) as the pole and the positive half of the \(x\)-axis as the polar axis. The parametric equations of line \(l\) are given by 
$$
\begin{cases}
x = t \\
y = t + 2
\end{cases} \quad (t \text{ is the parameter})
$$
The standard equation of curve \(C\) is \(x^{2}-4x+y^{2}-2y=0\). The polar coordinates of point \(P\) are \((2 \sqrt {2}, \dfrac {7\pi}{4})\).
1. Find the standard equation of the line \(l\) and the polar equation of the curve \(C\);
2. If line \(l\) is translated 2 units to the right to obtain line \(l'\), and \(l'\) intersects curve \(C\) at points \(A\) and \(B\), find the area of triangle \(PAB\).

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