
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
In today's society, "livestreaming sales" has become a new type of promotional method for businesses. Xiao Liang sells a daily commodity in the live broadcast room with a cost price of $10$ yuan per item. After investigation, it was found that the daily sales volume $y$ (items) of the commodity and the unit price $x$ (yuan) satisfy a linear function relationship, as shown in the table below:

| Unit Price $x$ (yuan) | $20$ | $25$ | $30$ |
|-----------------------|------|------|------|
| Sales Volume $y$ (items) | $200$ | $150$ | $100$ |

$(1)$ Find the functional relationship between $y$ and $x$;  
$(2)$ The business wants to make a profit of $2160$ yuan per day while minimizing inventory. What should be the unit price set at?

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