
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
Xiaogang went to the supermarket to buy paintbrushes. He spent $60$ yuan on a certain number of type $A$ paintbrushes for the first time. The supermarket recommended type $B$ paintbrushes for the second time, which were $2$ yuan more expensive per unit than type $A$ paintbrushes. He then spent $100$ yuan to buy the same number of type $B$ paintbrushes.<br/>$(1)$ What is the unit price of type $B$ paintbrushes at the supermarket?<br/>$(2)$ After using both types of paintbrushes, Xiaogang decided to use type $B$ paintbrushes in the future. However, he felt that the price was slightly expensive. After communicating with the supermarket, the supermarket provided the following discount plan: if you buy no more than $20$ brushes at a time, each type $B$ paintbrush will be discounted by $10\%$; if you buy more than $20 brushes at a time, the first $20$ brushes will be discounted by $10\%, and the remaining brushes will be discounted by $20\%$. Let $x$ be the number of type $B$ paintbrushes Xiaogang buys, and $y$ be the cost. Write the function relationship of $y$ with respect to $x$.<br/>$(3)$ Under the discount plan in $(2)$, if Xiaogang plans to buy type $B$ paintbrushes for $270$ yuan, how many type $B$ paintbrushes can he buy?

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