
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 the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with semi-major axis $a$ and semi-minor axis $b$, its semi-focal distance is $c$. If the line $y = 2x$ intersects the ellipse at a point with an x-coordinate of $c$, determine the eccentricity of the ellipse.

A) $1 - \frac{\sqrt{2}}{2}$
B) $\sqrt{2} - \frac{1}{2}$
C) $\sqrt{2} - 1$
D) $\sqrt{3} - 1$

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