
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 brand of rice is well-known far and wide and deeply loved by consumers. A supermarket purchases a batch of this rice every day at a cost of $4 per kilogram and sells it at a price not lower than the cost but not exceeding $7 per kilogram. When the selling price is $5 per kilogram, 950 kilograms of rice are sold every day; when the selling price is $6 per kilogram, 900 kilograms of rice are sold every day. An analysis of the sales data reveals that the quantity of rice sold per day $y$ (in kilograms) and the selling price per kilogram $x$ (in dollars) satisfy a linear function relationship.
- $(1)$ Please write down the function relationship between $y$ and $x$ directly.
- $(2)$ At what price per kilogram should the supermarket set the selling price so that the profit from selling the rice reaches $1800?
- $(3)$ At what price per kilogram should the selling price be set to maximize the daily profit? What is the maximum profit?

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