
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 the curve $C_1$ is
$$
\begin{cases}
x = 3 + \cos\theta \\
y = 4 + \sin\theta
\end{cases}
$$
where $\theta$ is the parameter. A polar coordinate system is established with the origin as the pole and the positive x-axis as the polar axis. The curve $C_2$ is given by the equation $\rho(\sin\theta - k\cos\theta) = 3$, where $k$ is a real number.
1. Find the general (Cartesian) equation of curve $C_1$ and the Cartesian equation of curve $C_2$.
2. If point $P$ lies on curve $C_2$ and a tangent line from point $P$ to $C_1$ has the minimum length of $2\sqrt{2}$, find the value of the real number $k$.

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