
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 the line $mx - y + m + 2 = 0$ intersects with circle $C\_1$: $(x + 1)^2 + (y - 2)^2 = 1$ at points $A$ and $B$, and point $P$ is a moving point on circle $C\_2$: $(x - 3)^2 + y^2 = 5$. Determine the maximum area of $\triangle PAB$.

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