
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 an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ that passes through point $(0, 4)$, with an eccentricity of $\frac{3}{5}$.

$(1)$ Find the equation of ellipse $C$;

$(2)$ Find the coordinates of the midpoint of the line segment cut by $C$ on the line that passes through $(3, 0)$ and has a slope of $\frac{4}{5}$.

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