
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
The 6th Travel and Tourism Conference of Hebei Province in 2023 was held in Handan. Two types of travel souvenirs, A and B, were issued for this event. A certain store is preparing to purchase 300 souvenirs. It is known that purchasing 40 type A souvenirs and 30 type B souvenirs costs 5000 yuan, and purchasing 10 type A souvenirs and 50 type B souvenirs costs 3800 yuan. The selling price of type A souvenirs is 120 yuan per item, and the selling price of type B souvenirs is 80 yuan per item.  
$(1)$ Find the cost price per item for type A and type B souvenirs, respectively.  
$(2)$ If the quantity of type B souvenirs is not less than 3 times the quantity of type A souvenirs, and the profit is not less than 7400 yuan. Let the profit be $w$ yuan. Calculate and determine the maximum profit for the store.  
$(3)$ If the selling price of type A souvenirs is reduced by $5m$ yuan per item, where $4 < m < 8$, and the selling price of type B souvenirs remains the same, under the conditions of $(2)$, the store sells these 300 souvenirs and obtains a maximum profit of 5720 yuan. Determine the value of $m$.

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