
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
On the eve of Teachers' Day, a flower shop purchased a batch of fresh flower gift boxes at a cost of $50$ yuan per piece. The Price Bureau requires that the profit margin obtained from selling this fresh flower gift box should not exceed $52\%$. An analysis of the sales of fresh flower gift boxes during the Teachers' Day period revealed that the daily sales volume $y$ (pieces) approximately satisfies a linear function relationship with the selling price $x$ (yuan per piece) (where $x$ is an integer). The data is as follows:

| Selling Price $x$ (yuan per piece) | $\ldots$ | $60$ | $70$ | $75$ | $\ldots$ |
|------------------------------------|----------|------|------|------|----------|
| Daily Sales Volume $y$ (pieces)    | $\ldots$ | $240$| $180$| $150$| $\ldots$ |

$(1)$ Find the functional relationship between $y$ and $x$;  
$(2)$ Determine the selling price at which the flower shop maximizes its daily profit from selling this fresh flower gift box, and calculate the maximum profit;  
$(3)$ The flower shop promises to donate $n$ yuan ($n < 2$) to the "Hope Project" for every fresh flower gift box sold. If the daily profit after deducting the donation increases with the selling price $x$, directly write down the range of values for $n$ as ____.

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