
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 equation of ellipse $C$ is $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, and the distance from its right focus $F$ to three vertices of the ellipse $C$ forms an arithmetic sequence with a common difference of $\sqrt{3}$. The sum of the three terms of this sequence is $6$.

1. Find the equation of ellipse $C$.
2. If line $AB$ intersects with ellipse $C$ at point $A$ and $B$ (with $A$ in the first quadrant), and satisfies $2\overrightarrow{OA}+ \overrightarrow{OB}=λ\overrightarrow{OF}$, find the equation of line $AB$ when the area of $\triangle OAB$ is the largest.

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