
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
Shao Yun Middle School plans to purchase two models of paints, $A$ and $B$, for the painting group. If they buy 1 box of paint of model $A$ and 2 boxes of paint of model $B$, it will cost $56$ yuan. If they buy 2 boxes of paint of model $A$ and 1 box of paint of model $B$, it will cost $64$ yuan. <br/> $(1)$ Find the price of each box of paint of model $A$ and model $B$. <br/> $(2)$ Shao Yun Middle School decides to purchase a total of $200$ boxes of the two models of paints, with a total cost not exceeding $3920$ yuan. How many boxes of paint of model $A$ can the school purchase at most?

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