
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 a Cartesian coordinate system, the parametric equation for curve $C_1$ is 
$$
\left\{ \begin{aligned} 
x &= 2\cos\alpha, \\
y &= \sqrt{2}\sin\alpha 
\end{aligned} \right.
$$
with $\alpha$ as the parameter. Using the origin as the pole, the positive half of the $x$-axis as the polar axis, and the same unit length as the Cartesian coordinate system, establish a polar coordinate system. The polar equation for curve $C_2$ is $\rho = \cos\theta$.

(1) Find the general equation for curve $C_1$ and the Cartesian coordinate equation for curve $C_2$;

(2) If $P$ and $Q$ are any points on the curves $C_1$ and $C_2$, respectively, find the minimum value of $|PQ|$.

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