
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 $(xOy)$, curve $C\_1$: $\begin{cases} x=\cos \alpha \\ y=\sin \alpha \end{cases} (\alpha$ is the parameter$)$ is given. Curve $C\_2$ is obtained by doubling the abscissa of all points on curve $C\_1$, and curve $C\_3$ is obtained by shifting curve $C\_1$ one unit upward. Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis.
(I) Find the ordinary equation of curve $C\_2$ and the polar coordinate equation of curve $C\_3$;
(II) If point $P$ is any point on curve $C\_2$ and point $Q$ is any point on curve $C\_3$, find the maximum value of $|PQ|$.

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