
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
Consider a hyperbola given by the equation $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$. Its left focus $F_1(-c, 0)$ intersects the circle $x^{2} + y^{2} = a^{2}$ at point $E$. The line segment $F_1E$ is extended to intersect the parabola $y^{2} = 4cx$ at point $P$. If $E$ is the midpoint of the line segment $F_1P$, determine the eccentricity of the hyperbola.

A) $\frac{1 + \sqrt{5}}{2}$
B) $\frac{1 + \sqrt{3}}{2}$
C) $\frac{3 + \sqrt{5}}{2}$
D) $\frac{\sqrt{5}}{2}$

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