
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
A certain city encourages residents to save electricity by using a tiered billing method to calculate the electricity bills for each household on a monthly basis. When the monthly electricity consumption does not exceed $200$ kilowatt-hours, the billing rate is $0.6$ yuan per kilowatt-hour. When the monthly electricity consumption exceeds $200$ kilowatt-hours, the first $200$ kilowatt-hours are still billed at $0.6$ yuan per kilowatt-hour, and the excess is billed at $0.65$ yuan per kilowatt-hour. Let $x$ represent the monthly electricity consumption in kilowatt-hours for each household, and $y$ represent the amount of money they need to pay.

$(1)$ When the monthly electricity consumption is not more than $200$ kilowatt-hours, the function relationship between $y$ and $x$ is ______; when the monthly electricity consumption exceeds $200$ kilowatt-hours, the function relationship between $y$ and $x$ is ______.

$(2)$ Xiaoxin's household consumed $160$ kilowatt-hours of electricity in October. Calculate how much money they need to pay for electricity in October.

$(3)$ Xiaoming's household paid $146$ yuan for electricity in October. Calculate how many kilowatt-hours of electricity they consumed in October.

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