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

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