
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, $F$ is the left focus of the ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$, and $A$, $B$ are respectively the left and right vertices of $C$. Point $P$ is on $C$, and $PF \perpendicular x$-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 $OE$, then calculate the eccentricity of $C$.

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