
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
In the Cartesian coordinate system xOy, the parametric equation of curve C₁ is given by:

$$
\begin{cases}
x=\cos \phi \\
y=\sin \phi
\end{cases}
$$

($\phi$ is the parameter), and the parametric equation of curve C₂ is:

$$
\begin{cases}
x=a\cos \phi \\
y=b\sin \phi
\end{cases}
$$

(with $a > b > 0$, $\phi$ being the parameter). In the polar coordinate system with O as the pole and the positive x-axis as the polar axis, the ray l: $\theta = \alpha$ intersects both C₁ and C₂ at one point each. When $\alpha = 0$, the distance between these two points is 2. When $\alpha = \frac{\pi}{2}$, these two points coincide.

(I) Specify what kind of curves C₁ and C₂ are, and determine the values of a and b;
(II) Suppose that when $\alpha = \frac{\pi}{4}$, the intersection points of l with C₁ and C₂ are A₁ and B₁, respectively, and when $\alpha = -\frac{\pi}{4}$, the intersection points with C₁ and C₂ are A₂ and B₂, respectively. Find the area of the quadrilateral A₁A₂B₂B₁.

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