
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 $\begin{cases} x=\cos \alpha \\ y=\sqrt{3}\sin \alpha \end{cases}(\alpha$ is the parameter$)$, and the polar coordinate equation of curve $C\_2$ with origin $O$ as the pole and positive semi-axis of $x$ as the polar axis is given by $\rho \sin (\theta + \frac{\pi}{4}) = 2\sqrt{2}$.

1. Find the Cartesian equation of curve $C\_1$ and the rectangular coordinate equation of curve $C\_2$.
2. Let $P$ be a moving point on curve $C\_1$, find the minimum distance from point $P$ to a point on curve $C\_2$.

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