
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 $C$ are $$x=2\cos\theta,\ y=\sin\theta$$, where $\theta$ is the parameter. In the polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equations of curves $C_1$ and $C_2$ are $\rho=8\cos\theta$ and $\rho=-8\sin\theta$, respectively.

1. Find the Cartesian equation of curve $C$ and the Cartesian equations of curves $C_1$ and $C_2$.
2. Let $A$ and $B$ be the intersection points of curves $C_1$ and $C_2$, $M$ be the midpoint of $AB$, and $N$ be a moving point on curve $C$. Find the minimum distance from the midpoint $P$ of $MN$ to the line $l$: $$x=2t,\ y=1+t$$, where $t$ is the parameter.

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