
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
A cubic parabola's inflection point is the center of a circle; this circle is internally tangent to the parabola at the points $A(\sqrt{2}, \sqrt{2})$ and $B(-\sqrt{2}, -\sqrt{2})$ and intersects it at two more points. The tangents drawn at points $A$ and $B$ are parallel to each other. If one of the intersection points' abscissa is $x_{1} < 0$, determine the area enclosed by the two curves between the limits of $x_{2} = \sqrt{2}$ and $x_{1}$.

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