
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 three liquids with densities in the ratio \(\rho_{1}: \rho_{2}: \rho_{3} = 6:3:2\). A mixture is prepared from these liquids such that the amount of the first liquid is at least 3.5 times more than the second liquid. In what proportion should the masses of these liquids be taken so that after mixing, the density of the mixture is equal to the arithmetic mean of the densities of the original liquids?

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