
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 shopping mall is preparing to purchase a new batch of goods from suppliers before "Double 11". It is known that purchasing $1$ unit of type A goods and $2$ units of type B goods costs a total of $320$ yuan, and purchasing $3$ units of type A goods and $2$ units of type B goods costs a total of $520$ yuan. <br/>$(1)$ Find the cost price per unit of type A and type B goods, respectively. <br/>$(2)$ If the selling price of type A goods is $120$ yuan per unit and the selling price of type B goods is $140$ yuan per unit, the shopping mall plans to purchase a total of $50$ units of type A and type B goods. After selling all these goods, the total profit should be no less than $1350$ yuan and no more than $1375$ yuan. If the shopping mall purchases $m$ units of type A goods, what are the possible purchasing plans? <br/>$(3)$ Based on the sales situation in previous years, the shopping mall plans to sell all 46 units of type A and type B goods at the selling prices mentioned in $(2)$ on "Double 11". However, due to the continuous influence of cold air caused by the La Niña phenomenon, rain and snow weather appeared on "Double 11", resulting in not all 46 units of goods being sold. The actual total profit from selling the two types of goods is $1220$ yuan. Therefore, how many units of type B goods were sold by the shopping mall on "Double 11" at least?

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