
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 ellipse $C:\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1(a>b>0)$ pass through the point $\left(0,4\right)$, with eccentricity $\frac{3}{5}$.<br/>$(1)$ Find the equation of $C$;<br/>$(2)$ If a line $l$ passing through the point $\left(3,0\right)$ with a slope of $\frac{4}{5}$ intersects the ellipse $C$ at points $A$ and $B$, find the length of the chord $|AB|$.

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