
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 two lines C<sub>1</sub> $$\begin{cases} \overset{x=1+t\cos\alpha }{y=t\sin\alpha }\end{cases}$$ (where $t$ is a parameter), C<sub>2</sub> $$\begin{cases} \overset{x=\cos\theta }{y=\sin\theta }\end{cases}$$ (where $\theta$ is a parameter),  
(Ⅰ) When $\alpha= \frac {\pi}{3}$, find the coordinates of the intersection point of C<sub>1</sub> and C<sub>2</sub>;  
(Ⅱ) Draw a perpendicular line from the origin O to C<sub>1</sub>, with the foot of the perpendicular being A, and P being the midpoint of OA. When $\alpha$ changes, find the parametric equation of the trajectory of point P, and identify the curve it represents.

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