
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 for curve C<sub>1</sub> is given by
$$
\begin{cases}
x = 2 + t\cos a \\
y = 1 + t\sin a
\end{cases}
$$
where $t$ is the parameter and $0 \leq a < \pi$. In the polar coordinate system with the origin O as the pole and the positive x-axis as the polar axis, the polar equation for curve C<sub>2</sub> is
$$
\rho^2 = \frac{2}{1 + \cos^2 \theta}.
$$

(1) Find the Cartesian equation for curve C<sub>1</sub> and the Cartesian equation for curve C<sub>2</sub>.
(2) When $\alpha = \frac{\pi}{4}$, curves C<sub>1</sub> and C<sub>2</sub> intersect at points M and N. Find the Cartesian equation of the circle with the segment MN as its diameter.

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