
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 rectangular coordinate system $(xOy)$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar coordinate equation of circle $C$ is $ρ=2 \sqrt {2}\sin (θ- \frac {π}{4})$, and the parametric equation of line $l$ is $ \begin{cases} x=- \frac { \sqrt {2}}{3}t \\ y=-1+ \frac { \sqrt {2}}{4}t\\end{cases}$, where line $l$ intersects circle $C$ at points $A$ and $B$. $P$ is any point on circle $C$ that is different from $A$ and $B$.

1. Find the rectangular coordinate equation of circle $C$.
2. Find the maximum area of $\triangle PAB$.

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