
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 rectangular coordinate system $(xOy)$, the parametric equations of curve $C\_1$ are given by $\begin{cases} x=2\cos \alpha \\ y=\sqrt{2}\sin \alpha \end{cases}$ ($\alpha$ is a parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis. The polar equation of curve $C\_2$ is given by $\rho \sin (θ+\frac{\pi}{4}) = 3\sqrt{2}$.

1. Find the Cartesian equation of curve $C\_1$ and the polar equation of curve $C\_2$.
2. Let $P\_1$ and $P\_2$ be two moving points on curves $C\_1$ and $C\_2$, respectively. Find the minimum value of line segment $P\_1P\_2$.

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