
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
Given lines $C\_1 \begin{cases} x=1+t\cos \alpha \\ y=t\sin \alpha \end{cases} (t\text{ is a parameter})$ and $C\_2 \begin{cases} x=\cos \theta \\ y=\sin \theta \end{cases} (\theta\text{ is a parameter})$,
(I) Find the coordinates of the intersection point(s) of $C\_1$ and $C\_2$ when $\alpha= \dfrac {\pi}{3}$;
(II) Draw a perpendicular line to $C\_1$ passing through the coordinate origin $O$, with the foot of the perpendicular being $A$, and $P$ as the midpoint of $OA$. Determine the parametric equation of the trajectory of point $P$ as $\alpha$ varies, and identify the type of curve it represents.

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