
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 \( C \):
\[ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad (a>b>0), \]

which passes through the point \( P\left(3, \frac{16}{5}\right) \) and has an eccentricity of \( \frac{3}{5} \). Draw a line \( l \) with slope \( k \) passing through the right focus of the ellipse \( C \), intersecting the ellipse at points \( A \) and \( B \). Let the slopes of \( PA \) and \( PB \) be \( k_{1} \) and \( k_{2} \) respectively.
(1) Find the standard equation of the ellipse.
(2) If \( k_{1} + k_{2} = 0 \), find the real number \( k \).

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