
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 equations of curve $C_{1}$ are given by $\begin{cases}x=2t-1 \\ y=-4t-2\end{cases}$ $(t$ is the parameter$)$, and in the polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equation of curve $C_{2}$ is $\rho= \frac{2}{1-\cos \theta}$.

(1) Write the Cartesian equation of curve $C_{2}$;

(2) Let $M_{1}$ be a point on curve $C_{1}$, and $M_{2}$ be a point on curve $C_{2}$. Find the minimum value of $|M_{1}M_{2}|$.

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