
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 $xOy$, curve $C_1$ is described by the parametric equations $\begin{cases} x= \sqrt{3}\cos \alpha, \\ y=\sin \alpha, \end{cases}$ (with $\alpha$ as the parameter). In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, curve $C_2$ has the polar equation $\rho\sin \left( \theta+ \dfrac{\pi}{4} \right)=2\sqrt{2}$.

(1) Write down the standard equation of $C_1$ and the Cartesian coordinate equation of $C_2$.

(2) Suppose point $P$ lies on $C_1$ and point $Q$ on $C_2$. Find the minimum value of $|PQ|$ and the Cartesian coordinates of $P$ at this minimum.

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