
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 city, residents are charged for water usage in a tiered manner. If the monthly water usage of a household does not exceed $5$ tons, the charge is $2$ yuan per ton; if it exceeds $5$ tons, the excess part is charged at $3.5$ yuan per ton. Let $x$ be the monthly water usage in tons of a household, and $y$ be the amount of money to be paid for water.<br/>$(1)$ Write down the functional relationship between $y$ and $x$ when the monthly water usage exceeds $5$ tons;<br/>$(2)$ If a resident paid $17$ yuan for water in a certain month, how many tons of water did the resident use?

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