
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 an ellipse $C$ with its center at the origin and the foci on the $x$-axis, the distance between the foci is $2$, and the length of the major axis is $\sqrt{2}$ times the length of the minor axis.

$(1)$ Find the standard equation of ellipse $C$.

$(2)$ Let $P(2,0)$ be a point, and a line $l$ with slope $k$ passing through the left focus $F$ of the ellipse $C$ intersects it at points $A$ and $B$. If the area of triangle $ABP$ is $\frac{\sqrt{10}}{2}$, find the equation of line $l$.

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