
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
Given a circle $C: (x+3)^2+(y-6)^2=36$, and a line $l$ passing through point $M(0, 3)$ divides circle $C$ into two parts, making the absolute value of the difference in areas of these two parts as large as possible.  
(Ⅰ) Find the equation of line $l$;  
(Ⅱ) Suppose line $l$ intersects circle $C$ at points $A$ and $B$, and point $P$ is another point on circle $C$ different from $A$ and $B$. Find the maximum value of the area of $\triangle PAB$.

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