
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 implements a tiered water pricing system for water charges, as shown in the table below:

| Monthly Water Usage | Up to 12 tons | Over 12 tons but not exceeding 18 tons | Exceeding 18 tons |
|---------------------|---------------|---------------------------------------|-------------------|
| Charge Rate (元/ton) | $2.00$ | $2.50$ | $3.00$ |

$(1)$ If a household paid $45$ yuan for water in May, what was their water usage for that month?

$(2)$ In order to keep the monthly water bill between $20$ yuan and $30$ yuan, what range should the household's monthly water usage be within?

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