
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 that the short axis of the ellipse $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ is half the length of the long axis, the shortest distance from a point on the ellipse $C$ to the right focus $F$ is $2-\sqrt{3}$, and the line $l$: $y=x+m$ intersects the ellipse $C$ at two distinct points $A(x_{1},y_{1}),B(x_{2},y_{2})$.

1. Find the equation of the ellipse $C$;
2. If the area of $\triangle AOB$ is $1$, find the equation of the line $l$.

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