
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 points $A(-2,0), B(0,2)$, and a constant real number $k$, points $M$ and $N$ are two distinct points on the circle defined by $x^2 + y^2 + kx = 0$. Point $P$ is a moving point on the circle $x^2 + y^2 + kx = 0$. If points $M$ and $N$ are symmetric with respect to the line $x - y - 1 = 0$, find the maximum value of the area of triangle $\Delta PAB$.

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