
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 ellipse \(C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\) that passes through the point \(P\left(3, \frac{16}{5}\right)\), with an eccentricity of \(\frac{3}{5}\). From the right focus of ellipse \(C\), a line \(l\) with slope \(k\) intersects 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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