
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's water company, in order to encourage residents to conserve water, began implementing a monthly segmented billing method based on individual users starting from month $C$ in the year $2023$. Until the end of March $2023$, the original billing standard was used. The two billing standards are as follows:

| Original Billing Standard | New Monthly Segmented Billing Standard |
|---------------------------|----------------------------------------|
| $2$ yuan per ton | $(1)$ For users who use no more than $10$ tons of water per month (including $10$ tons), the charge is $1.8$ yuan per ton; $2$ yuan per ton for any additional usage. <br/> $(2)$ For users who use more than $10$ tons of water per month, the first $10$ tons are charged at $1.8$ yuan per ton, and any excess over $10$ tons is charged at $a$ yuan per ton $(a > 1.8)$.

$(1)$ Resident A used $20$ tons of water in both March and April. However, in April, they paid $8$ yuan more than in March. Find the value of $a$. <br/> $(2)$ If Resident A uses $x$ tons of water in May, and should pay $y$ yuan, find the relationship between $y$ and $x$.

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