
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 line $y=x+1$ intersects the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at points $A$ and $B$ where the asymptotes of the hyperbola intersect at the origin $O$, and point $A$ lies in the first quadrant, and if $|OA|=2|OB|$, determine the eccentricity of the hyperbola.

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