
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
An ellipse $E$ is given by the equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ $(a > b > 0)$, with point $O$ at the coordinate origin, point $A$ at $(a, 0)$, and point $B$ at $(0, b)$. Point $M$ lies on line segment $AB$ such that $|BM| = 2|MA|$, and the slope of line $OM$ is $\frac{1}{4}$.

(1) Find the eccentricity $e$ of ellipse $E$.

(2) $PQ$ is a diameter of circle $C$: $(x+2)^2 + (y-1)^2 = \frac{15}{2}$. If ellipse $E$ passes through points $P$ and $Q$, find the equation of ellipse $E$.

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