
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 tea shop purchased several boxes of type $A$ tea leaves for $4000$ yuan and several boxes of type $B$ tea leaves for $8400$ yuan. The number of boxes of type $B$ tea leaves purchased was $10$ more than the number of boxes of type $A$ tea leaves purchased. The cost of each box of type $B$ tea leaves was $1.4$ times the cost of each box of type $A$ tea leaves.  
$(Ⅰ)$ What is the cost per box for type $A$ and type $B$ tea leaves, respectively?  
$(Ⅱ)$ After selling all the tea leaves purchased in the first round, the shop bought a total of $100$ boxes of type $A$ and type $B$ tea leaves in the second round (at the same cost). The selling price for each box of type $A$ tea leaves was $300$ yuan, and for each box of type $B$ tea leaves was $400$ yuan. After selling half of each type of tea leaves, to celebrate New Year's Day, both types of tea leaves were sold at a 30% discount. After selling all the tea leaves, the profit from the second round of purchases was $5800 yuan (ignoring other factors). Find out how many boxes of type $A$ and type $B$ tea leaves were purchased this time.

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