
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 Cartesian coordinate system $xOy$, the parametric equation of the circle $⊙O$ is $\begin{cases} x=\cos \theta \\ y=\sin \theta \end{cases}$, with $\theta$ as the parameter. A line $l$ passing through the point $(0, -\sqrt{2})$ with an inclination angle of $\alpha$ intersects the circle $⊙O$ at points $A$ and $B$.

$(1)$ Find the range of values for $\alpha$;  
$(2)$ Find the parametric equation of the trajectory of the midpoint $P$ of $AB$.

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