
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 the equation of the ellipse $E$ is $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, point $O$ is the coordinate origin, point $A$ has coordinates $(a,0)$, point $B$ has coordinates $(0,b)$, and point $M$ is on the line segment $AB$ and satisfies $|BM|=2|AM|$. The slope of the line $OM$ is $\frac{\sqrt{5}}{10}$.
(1) Find the eccentricity of the ellipse;
(2) Let point $C$ have coordinates $(-a,0)$, $N$ is the midpoint of the line segment $BC$, and the $y$-coordinate of the point symmetric to point $N$ about the line $AB$ is $\frac{13}{2}$. Find the equation of ellipse $E$.

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