
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 equation of line $l$ is $\begin{cases} x=1+ \frac{\sqrt{2}}{2}t \\ y= \frac{\sqrt{2}}{2}t \end{cases}$, where $t$ is the parameter. In the polar coordinate system with the origin $O$ as the pole and the $x$-axis as the polar axis, the polar equation of curve $C$ is $\rho=4\sin\theta$.  
(1) Find the general equation of line $l$ and the Cartesian coordinate equation of curve $C$;  
(2) Let $M$ be a moving point on curve $C$, and let the midpoint of $OM$ be $P$. Find the minimum distance from point $P$ to line $l$.

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