
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 clothing store intends to purchase a total of 100 pieces of two types of children's clothing, $A$ and $B$. Among them, there are 2 pieces of type $A$ and 1 piece of type $B$, with a total purchase price of 260 yuan; there are 1 piece of type $A$ and 3 pieces of type $B$, with a total purchase price of 380 yuan. The selling price of each piece of type $A$ is 120 yuan, and the selling price of each piece of type $B$ is 150 yuan.  
$(1)$ Find the purchase price of each piece of the two types of children's clothing.  
$(2)$ If 50 pieces of each type of children's clothing are purchased, find the profit after selling all these clothes.  
$(3)$ Due to financial constraints, the actual purchase quantity of type $A$ children's clothing is not less than twice the quantity of type $B$ children's clothing. In this case, what is the maximum number of pieces of type $B$ children's clothing that can be purchased? Also, find the profit after selling all these clothes.

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