
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
Eating zongzi during the Dragon Boat Festival is a traditional custom in China. During this year's Dragon Boat Festival, a supermarket sells two different types of zongzi, $A$ and $B$. It is known that buying 4 zongzi of type $A$ and 5 zongzi of type $B$ costs a total of $35$ yuan; buying 2 zongzi of type $A$ and 3 zongzi of type $B$ costs a total of $19$ yuan.  
$(1)$ Find the price of each zongzi of type $A$ and type $B$.  
$(2)$ A school plans to purchase a total of 1000 zongzi of types $A$ and $B$ with a budget of no more than 4000 yuan. How many zongzi of type $A$ can be purchased at most?

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