
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 certain new product was purchased by Hefei Department Store at a cost of $120$ yuan per item. During the trial sales phase in May, it was found that there exists a relationship between the selling price (in yuan) per item and the daily sales volume (in items) of the product as shown in the table below:

| Selling Price per Item (yuan) | $130$ | $135$ | $140$ | $\ldots$ | $180$ | $\ldots$ |
|-------------------------------|-------|-------|-------|---------|-------|---------|
| Daily Sales Volume (items)    | $70$  | $65$  | $60$  | $\ldots$ | $a$   | $\ldots$ |

$(1)$ Please observe the changing pattern of the data in the table and fill in the value of $a$ as ______.

$(2)$ If the department store wants to achieve a daily profit of $1600$ yuan for this product, what should be the selling price per item?

$(3)$ Salesperson Xiao Li found that the profit from selling $m$ items and $n$ items of this product is the same, where $m\neq n$. Write down the relationship between $m$ and $n$ directly.

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