
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 $O$ is the origin of coordinates, $F_1$ and $F_2$ are respectively the left and right foci of the hyperbola $C$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, $A$ is the left vertex of $C$, $P$ is a point on $C$, and $PF_1$ is perpendicular to the $x$-axis. A line through point $A$ intersects the line segment $PF_1$ at point $M$ and intersects the $y$-axis at point $E$. If the line $F_2M$ intersects the $y$-axis at point $N$, and $OE = 2ON$, determine the eccentricity of $C$.

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