
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
"In the worldly flavor, it touches the hearts of ordinary people." Street economy and small shop economy are important sources of employment. A certain operator purchased two types of toys, type $A$ and type $B$. It is known that the quantity of type $A$ toys purchased for $520$ yuan is $30$ more than the quantity of type $B$ toys purchased for $175$ yuan, and the unit price of type $A$ toys is $1.6$ times the unit price of type $B$ toys. <br/>$(1)$ Find the unit price of each type of toy. According to the question, two students, A and B, respectively set up the following equations: <br/>A: $\frac{520}{1.6x}=\frac{175}{x}+30$, solving gives $x=5$, which is a valid solution. <br/>B: $\frac{520}{x}=1.6\times \frac{175}{x-30}$, solving gives $x=65$, which is a valid solution. Then in the equation set up by A, $x$ represents ______, and in the equation set up by B, $x$ represents ______. <br/>$(2)$ If the operator is preparing to purchase these two types of toys again at the original unit price for a total of $200$ toys with $1350$ yuan, how many type $A$ toys can be purchased at most?

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