
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 $xOy$, a curve $C_1$ is given by the parametric equations: $$\begin{cases} x=-2+2\cos t\\ y=1+2\sin t\end{cases}$$ (where $t$ is the parameter). In the polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, another curve $C_2$ is given by the equation $4\rho\cos\theta-\rho\sin\theta-1=0$.
(1) Find the Cartesian equation for curve $C_1$ and the polar equation for curve $C_2$.
(2) If point $P$ is on curve $C_1$ and point $Q$ is on curve $C_2$, find the minimum value of the distance $|PQ|$.

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