
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 that points \( A \), \( B \), and \( C \) lie on the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) and point \( F(3,0) \), if \(\overrightarrow{F A}+\overrightarrow{F B}+\overrightarrow{F C}=\mathbf{0}\), find \(|\overrightarrow{F A}|+|\overrightarrow{F B}|+|\overrightarrow{F C}|\).

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