
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
Let the equation of ellipse $E$ be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$, point $O$ is the origin, point $A$ has coordinates $(a, 0)$, point $B$ has coordinates $(0, b)$, point $M$ is on segment $AB$, satisfying $|BM| = 2|AM|$, the slope of line $OM$ is $\frac{\sqrt{5}}{10}$.

$(1)$ Find the eccentricity of ellipse $E$;

$(2)$ Suppose point $C$ has coordinates $(-a, 0)$, $N$ is the midpoint of segment $BC$, the ordinate of the symmetric point of point $N$ with respect to 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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