
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
A coordinate system and parametric equations are given in the plane rectangular coordinate system $xOy$. The parametric equations of the curve $C\_1$ are $\begin{cases} x=\sqrt{2}\sin(\alpha + \frac{\pi}{4}) \\ y=\sin(2\alpha) + 1 \end{cases}$, where $\alpha$ is the parameter. Establish a polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C\_2$ is $\rho^2 = 4\rho\sin\theta - 3$.

1. Find the Cartesian equation of the curve $C\_1$ and the polar coordinate equation of the curve $C\_2$.
2. Find the minimum distance between a point on the curve $C\_1$ and a point on the curve $C\_2$.

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