
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
The fixed costs of a water sales department are $200$ yuan per day, including rent and staff wages. The cost price of each barrel of water is $5$ yuan, and the selling price of each barrel of water is $8$ yuan. Let $x$ (unit: barrels) represent the daily sales quantity, and $y$ (unit: yuan) represent the daily profit. (Profit $=$ Total sales revenue $-$ Fixed costs $-$ Cost of sold water). <br/> $(1)$ Find the functional relationship between $y$ and $x$; <br/> $(2)$ If the sales quantity on a certain day is $240$ barrels, what is the profit on that day in yuan? <br/> $(3)$ If the profit on a certain day is $760$ yuan, how many barrels of water were sold on that day?

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