
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
Given that the pole of the polar coordinate system coincides with the origin of the Cartesian coordinate system, and the polar axis coincides with the non-negative half-axis of $C = \frac{\pi}{2}$, with the same unit of length. The curve $C_{1}: \rho\cos \theta - 2\rho\sin \theta - 7 = 0$, and $C_{2}: \begin{cases} x=8\cos \theta \\ y=3\sin \theta \end{cases} (\theta \text{ as parameter})$.

$(1)$ Write the Cartesian coordinate equation of $C_{1}$ and the general equation of $C_{2}$;  
$(2)$ Given point $P(-4,4)$, $Q$ is a moving point on $C_{2}$, find the minimum distance from the midpoint $M$ of $PQ$ to the curve $C_{1}$.

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