
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 curve $C_1$ is $$\begin{cases} x=2\cos\alpha \\ y= \sqrt {2}\sin\alpha \end{cases}$$ ($\alpha$ is the parameter). Taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinate equation of curve $C_2$ is $\rho\sin\left(\theta+ \frac {\pi}{4}\right)=3 \sqrt {2}$.

(1) Find the general equation of curve $C_1$ and the Cartesian coordinate equation of curve $C_2$;  
(2) Let $P_1$ and $P_2$ be two moving points on curves $C_1$ and $C_2$, respectively. Find the minimum value of the line segment $P_1P_2$.

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