
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
The electricity price in a certain region last year was 0.80 yuan per kilowatt-hour, with an annual electricity consumption of \( a \) kilowatt-hours. This year, there is a plan to reduce the electricity price to between 0.55 yuan per kilowatt hour and 0.75 yuan per kilowatt hour, while the users expect the electricity price to be 0.40 yuan per kilowatt hour.

According to calculations, the increased electricity consumption after reducing the price is inversely proportional to the difference between the actual price and the users' expected price (the proportionality coefficient is \( k \)). The electricity cost in this region is 0.30 yuan per kilowatt hour.

(1) Write the functional relationship between the power department’s revenue \( y \) and the actual price \( x \) after reducing the price this year.
(2) Given that \( k = 0.2a \), what is the minimum price at which the power department’s revenue can still increase by at least 20% compared to last year?

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