
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 \( A_{1}, A_{2}, \ldots, A_{n} \) be fixed points, and \( k_{1}, k_{2}, \ldots, k_{n} \) be given numbers. Then the locus of points \( M \) such that the sum \( k_{1}\left|A_{1}M\right|^{2} + k_{2}\left|A_{2}M\right|^{2} + \ldots + k_{n}\left|A_{n}M\right|^{2} \) is constant will be:

a) A circle, a point, or an empty set if \( k_{1} + k_{2} + \ldots + k_{n} \neq 0 \);

b) A line, an empty set, or the entire plane if \( k_{1} + k_{2} + \ldots + k_{n} = 0 \).

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