
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 $xOy$, it is known that the center of the ellipse $E$ is at the origin, and its major axis is $8$ in length. One of the foci of the ellipse $E$ is the center of the circle $C$: $x^{2}+y^{2}-4x+2=0$.

$(1)$ Find the standard equation of the ellipse $E$;  
$(2)$ Let $P$ be a point on the left side of the $y$-axis on the ellipse $E$. Two lines $l_{1}$ and $l_{2}$ passing through $P$ have the product of their slopes equal to $\dfrac{1}{2}$. When both lines $l_{1}$ and $l_{2}$ are tangent to the circle $C$, find the coordinates of $P$.

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