
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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$). Construct a tangent line to the circle $x^2 + y^2 = a^2$ from the hyperbola's left focus $F(-c,0)$, with the point of tangency being $E$. Extend $FE$ to intersect the parabola $y^2 = 4cx$ at the point $P$. If $E$ is the midpoint of the line segment $FP$, find the eccentricity of the hyperbola.

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