
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

For given natural numbers \( k_{0} < k_{1} < k_{2} \), determine the minimum number of roots in the interval \([0, 2\pi)\) that the equation

\[ 
\sin k_{0} x + A_{1} \sin k_{1} x + A_{2} \sin k_{2} x = 0 
\]

can have, where \( A_{1}, A_{2} \in \mathbb{R} \).

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