
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 a Cartesian coordinate system, the parametric equations of curve C<sub>1</sub> are given as
$$
\begin{cases}
x = 2 + 2\cos\theta \\
y = 2\sin\theta
\end{cases}
$$
where $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $\theta$ is the parameter. If we establish a polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation of curve C<sub>2</sub> is $\theta = \frac{\pi}{4}$ with $\rho \in \mathbb{R}$.

(I) Find the general equation of curve C<sub>1</sub> and the Cartesian equation of curve C<sub>2</sub>.
(II) If curve C<sub>2</sub> is translated downward by m units (where m > 0) and the resulting curve has exactly two points in common with curve C<sub>1</sub>, determine the range of values for the real number m.

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