
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 the line $2x-y-2=0$ intersects the hyperbola $C:x^{2}-y^{2}=1$ at points $A(x_{1}, y_{1})$, $B(x_{2}, y_{2})$, and $P(x_{3}, y_{3})$ is a point on $C$, with $x_{1} \lt x_{3} \lt x_{2}$, $y_{1} \lt y_{3} \lt y_{2}$. Find the maximum area of triangle $\triangle PAB$.

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