
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 ancient city of Chaotian in Guangyuan City has a long history, dating back more than 2000 years. Due to its unique folk culture, it has attracted many tourists. To seize the business opportunity, a certain store decided to purchase two types of souvenirs, $A$ and $B$. It is known that purchasing 10 items of type $A$ and 5 items of type $B$ costs a total of $2000$ yuan; purchasing 5 items of type $A$ and 3 items of type $B$ costs a total of $1050$ yuan.<br/>$(1)$ How much does each item of type $A$ and type $B$ cost?<br/>$(2)$ If the store decides to spend $4000$ yuan to purchase these two types of souvenirs, with at least 12 items of each type, how many different purchasing plans are there for the store?

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