
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 $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $e=\frac{{\sqrt{3}}}{2}$, and the shortest distance from the focus to a point on the ellipse is $2-\sqrt{3}$. <br/>$(1)$ Find the standard equation of the ellipse. <br/>$(2)$ Let the line $l:y=kx+1$ intersect the ellipse at points $M$ and $N$. When $|MN|=\frac{{8\sqrt{2}}}{5}$, find the equation of the line $l$.

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