
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 that circle $M$ passes through $A(2,2)$, $B(6,0)$, and the center of the circle lies on the line $x-y-4=0$.
1. Find the equation of this circle.
2. Find the equation of a line that is perpendicular to the line $3x-y+5=0$ and tangent to the circle.
3. If point $P$ is any point on circle $M$, find the maximum area of triangle $\triangle ABP$.

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