
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 the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$ passes through point $A(0,4)$, and its eccentricity is $\dfrac {3}{5}$;  
$(1)$ Find the equation of ellipse $C$;  
$(2)$ Find the coordinates of the midpoint of the line segment intercepted by $C$ on the line passing through point $(3,0)$ with a slope of $\dfrac {4}{5}$.

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