
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
The center of an ellipse at the origin, with one of the foci being F1(0, c), is intersected by the line y = 3x - 2 creating a chord. If the x-coordinate of the midpoint of this chord is 1/2, find the eccentricity of this ellipse.

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