
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 $xOy$, the parametric equations of a line are given by $\begin{cases} x=2t \\ y=2 \sqrt {3}t+ \frac {1}{2}\end{cases}$ (where $t$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole, the non-negative semi-axis of $x$ as the polar axis, and the same unit length. The polar equation of circle $C$ is $\rho=2\sin\theta$.

(1) Find the Cartesian equation of circle $C_1$ and the general equation of line $l$.
(2) Suppose point $A$ moves on circle $C_1$, and the locus of the midpoint $P$ of segment $OA$ is curve $C_2$. The intersection points of curve $C_2$ and line $l$ are $M$ and $N$, with point $M$ having a greater $x$-coordinate than point $N$. Find the Cartesian coordinates of points $M$ and $N$.

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