
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 region, the electricity price is $0.8$ yuan per $(kW\cdot h)$, and the annual electricity consumption is $100$ million $(kW\cdot h)$. This year, the power department plans to reduce the electricity price to increase electricity consumption and revenue. The increase in electricity consumption after the price reduction is directly proportional to the square of the difference between the actual price and the original price, with a proportionality constant of $50$. The cost of electricity in this region is $0.5$ yuan per $(kW\cdot h)$.
1. Write the function relationship between the revenue $y$ and the actual price $x$;
2. What is the pattern of change in $y$ as $x$ changes?
3. What price should the power department set to maximize revenue?

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