
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 shopping mall purchases two types of toys, $A$ and $B$, at the World Expo. The unit price of toy $B$ is $25$ yuan more expensive than the unit price of toy $A$, and purchasing $2$ toys of type $B$ and $1$ toy of type $A$ costs a total of $200$ yuan.  
$(1)$ Find the unit prices of toys $A$ and $B$;  
$(2)$ If the shopping mall requires the quantity of toy $B$ to be twice the quantity of toy $A$, and the total amount spent on purchasing toys does not exceed $20000$ yuan, how many toys of type $A$ can the shopping mall purchase at most?

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