
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, $F$ is the left focus of the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$. $A$ and $B$ are respectively the left and right vertices of $C$. $P$ is a point on $C$, and $PF$ is perpendicular to the $x$-axis. A line through point $A$ intersects segment $PF$ at point $M$, and intersects the $y$-axis at point $E$. If line $BM$ passes through the midpoint of $OE$, calculate the eccentricity of $C$.

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