
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]


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A batch of greeting cards with a cost price of $2$ yuan is sold in a shopping mall. In marketing, it is found that there is a relationship between the daily selling price $x$ (in yuan) and the daily sales volume $y$ (in units) of this product as shown in the table below:

| $x$ / yuan | $3$ | $4$ | $5$ | $6$ |
|------------|----|----|----|----|
| $y$ / units | $20$ | $15$ | $12$ | $10$ |

$(1)$ Write the analytical expression of $y$ in terms of $x$.  
$(2)$ Let the daily profit from selling these greeting cards be $W$ (in yuan). Find the analytical expression of $W$ in terms of $x$. If the Price Bureau stipulates that the daily selling price of these greeting cards cannot exceed $10$ yuan per unit, determine the value of $x$ at which the maximum daily profit can be obtained, and calculate the maximum daily profit.

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