
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 last year was $0.8$ yuan$(/kW⋅h)$, with an annual electricity consumption of $akW⋅h$. This year, the plan is to lower the electricity price to between $0.55$ yuan$(/kW⋅h)$ and $0.75$ yuan$(/kW⋅h)$. However, users hope for a price of $0.4$ yuan$(/kW⋅h)$. Upon calculation, it was found that the increase in electricity consumption after lowering the price is inversely proportional to the difference between the actual price and the desired price (with a proportionality constant of $K$). The cost of electricity in this region is $0.3$ yuan$(/kW⋅h)$.
$(1)$ Write the function relationship $y$ between the power department's revenue and the actual price $x$ after lowering the price this year;
$(2)$ If $k=0.2a$, what is the lowest price that can still ensure that the power department's revenue increases by at least $20%$ compared to last year? (Note: Revenue $=$ Actual electricity consumption $(×($ Actual price $- $ Cost price $))$)

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