
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
With the advancement of the national rural revitalization policy, the agricultural and sideline products of Fenghuang Village are becoming more and more abundant. In order to increase the income of the villagers, a plan is made to price and sell a certain local specialty. They conducted a 4-day trial sales of this local specialty (costing $10$ yuan per bag), and recorded the daily sales volume $y$ (bags) and the selling price per bag $x$ (yuan) as follows:

| Time | Day 1 | Day 2 | Day 3 | Day 4 |
|------|-------|-------|-------|-------|
| $x/$ yuan | $15$ | $20$ | $25$ | $30$ |
| $y/$ bags | $25$ | $20$ | $15$ | $10$ |

If during the trial sales and normal sales period, the linear relationship between the daily sales volume $y$ and the selling price per bag $x$ remains the same, solve the following problems:

$(1)$ Find the functional relationship expression of the daily sales volume $y$ with respect to the selling price per bag $x$.

$(2)$ Help the villagers design at what price per bag should be set to maximize the daily profit of this local specialty, and calculate the maximum profit. (Profit $=$ Sales revenue $-$ Cost)

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