
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 government, in order to guide residents to use water reasonably, has decided to fully implement a tiered water pricing system, with residential units (one residential unit per household) as the principle unit of water usage.

| Tier | Tier 1 | Tier 2 | Tier 3 |
|------|--------|--------|--------|
| Monthly Water Usage (tons) | $(0,12]$ | $(12,16]$ | $(16,+\infty)$ |

To understand the distribution of monthly water usage among residents in the city, a sample of 10 households' monthly water usage (in tons) was obtained, and the statistics are shown in the table below:

| Household Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|------------------|---|---|---|---|---|---|---|---|---|----|
| Water Usage (tons) | 7 | 8 | 8 | 9 | 10 | 11 | 13 | 14 | 15 | 20 |

(1) If a resident uses 17 tons of water, how much money should they pay for water? If the water usage does not exceed 12 tons, the water fee is calculated at 4 yuan per ton; if the water usage exceeds 12 tons but does not exceed 16 tons, the excess over 12 tons is calculated at 5 yuan per ton; if the water usage exceeds 16 tons, the excess over 16 tons is calculated at 7 yuan per ton.

(2) Now, from these 10 households, 3 households are randomly selected. Find the distribution table and expectation of the number of households that fall into the second tier of water usage.

(3) Using the 10 sampled households as a sample to estimate the water usage of residents in the entire city, 10 households are randomly selected from the city one by one. If the number of households with water usage falling into the first tier is maximized, find the value of k.

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