
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 with its center at the origin $O$, its foci on the $x$-axis, and a line with a slope of $1$ passing through the right focus $F$ of the ellipse and intersecting the ellipse at points $A$ and $B$. If $\overrightarrow{OA}+\overrightarrow{OB}$ is collinear with $a=(3,-1)$, find the eccentricity of the ellipse.

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