
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]


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In the rectangular coordinate system $(xOy)$, the parametric equations of curve $C_1$ are given by $\begin{cases} x=2\cos ^{2}\alpha \\ y=\sin 2\alpha \end{cases} (\alpha \text{ is the parameter})$, and the polar coordinate equation of curve $C_2$ with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis is $\rho= \frac {1}{\sin \theta -\cos \theta }$.

(1) Find the Cartesian equation of curve $C_1$ and the polar coordinate equation of curve $C_2$.

(2) Find the minimum distance from any point $P$ on curve $C_1$ to curve $C_2$, and determine the coordinates of point $P$ in this case.

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