
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

In a certain city, all residents can be divided into three equal groups: poor, middle, and rich. The income of the poor group is \(x\) percent of the total income of all residents, the income of the middle group is \(4x\) percent, and the income of the rich is \(5x\) percent.

The city decided to introduce a tax on the income of the rich group, amounting to \(\left(\frac{x^{2}}{4}+x\right)\) percent of their income. The collected tax is distributed as follows: three-quarters of the collected sum goes to the poor, and one-quarter goes to the middle group. Calculate the incomes of the groups after the introduction of the taxes (as a percentage of the total income of all residents).

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