
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 coordinate origin, $F$ is the left focus of the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, $A$, $B$ are the left and right vertices of $C$ respectively. $P$ is a point on $C$, and $PF \perp x$ axis. The line $l$ passing through point $A$ intersects line segment $PF$ at point $M$, and intersects the $y$ axis at point $E$. If line $BM$ passes through the midpoint of $OE$, then the eccentricity of $C$ is _____.

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