
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 $(xOy)$, a curve $C_1$ is defined by the parametric equations: $\begin{cases} x=\cos \theta \\ y=\sin \theta \end{cases}$ where $\theta$ is the parameter. By stretching the coordinates of all points on $C_1$ horizontally by a factor of $\sqrt{2}$ and vertically by a factor of $2$, we obtain the curve $C_2$. In the same coordinate system, consider the line $l: \rho (\sqrt{2}\cos \theta + \sin \theta) = 4$.

(1) Write the polar equation of the curve $C_1$ and the parametric equations of the curve $C_2$;

(2) Find a point $P$ on the curve $C_2$ such that the distance from $P$ to the line $l$ is minimum, and compute this minimum distance.

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