
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 store deals with two types of products, A and B. The following information is available:<br/>Information $1$: The sum of the purchase prices of products A and B is $3$ yuan;<br/>Information $2$: The retail price of product A is $1$ yuan more than its purchase price, and the retail price of product B is $1$ yuan less than twice its purchase price;<br/>Information $3$: Buying 3 units of product A and 2 units of product B at their retail prices costs a total of $12$ yuan.<br/>Based on the above information, answer the following questions:<br/>$(1)$ Find the retail prices of products A and B;<br/>$(2)$ The store sells an average of 500 units of products A and B each day. After investigating, it was found that for every $0.1$ yuan decrease in the retail price of product A, the store can sell an additional 100 units of product A per day. The store decides to decrease the retail price of product A by $m\left(m \gt 0\right)$ yuan. Under the condition of not considering other factors, for what value of $m$ will the store's total profit from selling products A and B each day be $1000$ yuan?

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