
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
Point $P$ moves on the circle $O: x^{2}+y^{2}=4$. A perpendicular line segment $PD$ is drawn passing through point $P$ with $D$ as the foot of the perpendicular. $M$ is the midpoint of segment $PD$ (when point $P$ passes through the intersection points of the circle and the $x$-axis, point $M$ coincides with point $P$).  
$(1)$ Find the equation of the locus of point $M$.  
$(2)$ Through the point $({\sqrt{3},0})$, a line $l$ is drawn intersecting circle $O$ at points $A$ and $B$, and intersecting the locus of point $M$ at points $C$ and $D$. If $|AB|\cdot |CD|=\frac{{8\sqrt{10}}}{5}$, find the equation of line $l$.

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