
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
When a craft store sells a certain craft item at the marked price, it makes a profit of $45$ yuan per item. Moreover, the cost of purchasing $50$ items of the craft is the same as selling $40$ items of the craft at the marked price.<br/>$(1)$ What are the cost price and the marked price of each craft item?<br/>$(2)$ If each craft item is purchased at the cost price obtained in $(1)$ and sold at the marked price, the store can sell 100 of these craft items per day. If the price of each craft item is reduced by $1$, the store can sell an additional 4 items per day. How much should the price of each craft item be reduced to maximize the profit per day, and what is the maximum profit obtained?

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