
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 supermarket purchased a batch of fruit at a cost of $8$ yuan per kilogram. According to market research, the selling price $m($ yuan per kilogram$)$ of this fruit in the next $10$ days is related to the day $x$ by the function $m=\frac{1}{2}x+18(1\leqslant x\leqslant 10$, $x$ is an integer). Through analyzing the sales situation, it was found that the daily sales volume $y($kg$)$ is related to the day $x$ by a linear function. The table below shows three sets of corresponding values.<br/>

| Day $x$ | $\ldots $ | $2$ | $5$ | $9$ | $\ldots $ |
|---------|-----------|-----|-----|-----|-----------|
| Sales $y$/kg | $\ldots $ | $33$ | $30$ | $26$ | $\ldots $ |

$(1)$ Find the analytical expression of the function for $y$ in terms of $x$;<br/>
$(2)$ In these $10$ days, on which day is the profit from selling this fruit the highest, and what is the maximum profit in yuan?

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