
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]


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In a certain city, all residents can be divided into three equally sized groups: poor, middle, and rich. The poor group's income is $x$ percent of the total income of all city residents, the middle group's income is $3x$ percent, and the rich group's income is $6x$ percent.

The city decided to introduce an income tax for the rich, which is $\left(\frac{x^{2}}{5} + x\right)$ percent of their income. The collected tax amount is distributed as follows: two-thirds of the amount goes to the poor, and one-third goes to the middle group. Calculate the incomes of the groups after the tax is applied (as a percentage of the total income of all city residents).

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