
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
Calculate the line integral \(I=\int_{L}(xy - 1) \, dx + x^{2} y \, dy\) from point \(A(1, 0)\) to point \(B(0, 2)\): 
1. Along the line \(2x + y = 2\);
2. Along the arc of the parabola \(4x + y^{2} = 4\);
3. Along the arc of the ellipse \(x = \cos t, y = 2 \sin t\).

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