
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 the line $C_1$: $\begin{cases}x=1+t\cos \alpha, \\ y=t\sin \alpha,\end{cases}$ (where $t$ is a parameter), and the circle $C_2$: $\begin{cases}x=\cos \theta, \\ y=\sin \theta,\end{cases}$ (where $\theta$ is a parameter),

(1) When $\alpha= \frac{\pi}{3}$, find the coordinates of the intersection points of $C_1$ and $C_2$;

(2) Draw a perpendicular line from the origin $O$ to $C_1$, with the foot of the perpendicular being $A$, and $P$ being the midpoint of $OA$. When $\alpha$ varies, find the parametric equation of the trajectory of point $P$, and identify what curve it is.

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